Theorem rankuni 4698

Description: The rank of a union. Part of Exercise 4 of [Kunen] p. 107.

Assertion

Ref	Expression
rankuni	|- (rank` U.A) = U.(rank` A)

Proof of Theorem rankuni

Step	Hyp	Ref	Expression
1		unieq 2510	. . . . 5 |- (x = A -> U.x = U.A)
2	1	fveq2d 3728	. . . 4 |- (x = A -> (rank` U.x) = (rank` U.A))
3		fveq2 3724	. . . . 5 |- (x = A -> (rank` x) = (rank` A))
4	3	unieqd 2512	. . . 4 |- (x = A -> U.(rank` x) = U.(rank` A))
5	2, 4	eqeq12d 1489	. . 3 |- (x = A -> ((rank` U.x) = U.(rank` x) <-> (rank` U.A) = U.(rank` A)))
6		visset 1813	. . . . . . 7 |- x e. V
7	6	rankuni2 4690	. . . . . 6 |- (rank` U.x) = U_z e. x (rank` z)
8		fvex 3732	. . . . . . 7 |- (rank` z) e. V
9	8	dfiun2 2587	. . . . . 6 |- U_z e. x (rank` z) = U.{y | E.z e. x y = (rank` z)}
10	7, 9	eqtr 1495	. . . . 5 |- (rank` U.x) = U.{y | E.z e. x y = (rank` z)}
11		df-rex 1650	. . . . . . . 8 |- (E.z e. x y = (rank` z) <-> E.z(z e. x /\ y = (rank` z)))
12	6	rankel 4680	. . . . . . . . . . 11 |- (z e. x -> (rank` z) e. (rank` x))
13	12	anim1i 334	. . . . . . . . . 10 |- ((z e. x /\ y = (rank` z)) -> ((rank` z) e. (rank` x) /\ y = (rank` z)))
14	13	19.22i 1040	. . . . . . . . 9 |- (E.z(z e. x /\ y = (rank` z)) -> E.z((rank` z) e. (rank` x) /\ y = (rank` z)))
15		19.42v 1308	. . . . . . . . . 10 |- (E.z(y e. (rank` x) /\ y = (rank` z)) <-> (y e. (rank` x) /\ E.z y = (rank` z)))
16		eleq1 1534	. . . . . . . . . . . 12 |- (y = (rank` z) -> (y e. (rank` x) <-> (rank` z) e. (rank` x)))
17	16	pm5.32ri 646	. . . . . . . . . . 11 |- ((y e. (rank` x) /\ y = (rank` z)) <-> ((rank` z) e. (rank` x) /\ y = (rank` z)))
18	17	exbii 1051	. . . . . . . . . 10 |- (E.z(y e. (rank` x) /\ y = (rank` z)) <-> E.z((rank` z) e. (rank` x) /\ y = (rank` z)))
19		pm3.26 319	. . . . . . . . . . 11 |- ((y e. (rank` x) /\ E.z y = (rank` z)) -> y e. (rank` x))
20		rankon 4671	. . . . . . . . . . . . . . . 16 |- (rank` x) e. On
21	20	onel 3098	. . . . . . . . . . . . . . 15 |- (y e. (rank` x) -> y e. On)
22		rankr1id 4697	. . . . . . . . . . . . . . 15 |- (y e. On <-> (rank` (R1` y)) = y)
23	21, 22	sylib 198	. . . . . . . . . . . . . 14 |- (y e. (rank` x) -> (rank` (R1` y)) = y)
24	23	eqcomd 1480	. . . . . . . . . . . . 13 |- (y e. (rank` x) -> y = (rank` (R1` y)))
25		fvex 3732	. . . . . . . . . . . . . 14 |- (R1` y) e. V
26		fveq2 3724	. . . . . . . . . . . . . . 15 |- (z = (R1` y) -> (rank` z) = (rank` (R1` y)))
27	26	eqeq2d 1486	. . . . . . . . . . . . . 14 |- (z = (R1` y) -> (y = (rank` z) <-> y = (rank` (R1` y))))
28	25, 27	cla4ev 1869	. . . . . . . . . . . . 13 |- (y = (rank` (R1` y)) -> E.z y = (rank` z))
29	24, 28	syl 10	. . . . . . . . . . . 12 |- (y e. (rank` x) -> E.z y = (rank` z))
30	29	ancli 296	. . . . . . . . . . 11 |- (y e. (rank` x) -> (y e. (rank` x) /\ E.z y = (rank` z)))
31	19, 30	impbi 157	. . . . . . . . . 10 |- ((y e. (rank` x) /\ E.z y = (rank` z)) <-> y e. (rank` x))
32	15, 18, 31	3bitr3 181	. . . . . . . . 9 |- (E.z((rank` z) e. (rank` x) /\ y = (rank` z)) <-> y e. (rank` x))
33	14, 32	sylib 198	. . . . . . . 8 |- (E.z(z e. x /\ y = (rank` z)) -> y e. (rank` x))
34	11, 33	sylbi 199	. . . . . . 7 |- (E.z e. x y = (rank` z) -> y e. (rank` x))
35	34	abssi 2122	. . . . . 6 |- {y | E.z e. x y = (rank` z)} (_ (rank` x)
36		uniss 2521	. . . . . 6 |- ({y | E.z e. x y = (rank` z)} (_ (rank` x) -> U.{y | E.z e. x y = (rank` z)} (_ U.(rank` x))
37	35, 36	ax-mp 7	. . . . 5 |- U.{y | E.z e. x y = (rank` z)} (_ U.(rank` x)
38	10, 37	eqsstr 2091	. . . 4 |- (rank` U.x) (_ U.(rank` x)
39		pwuni 2757	. . . . . . . 8 |- x (_ P~U.x
40	6	uniex 2870	. . . . . . . . . 10 |- U.x e. V
41	40	pwex 2745	. . . . . . . . 9 |- P~U.x e. V
42	41	rankss 4688	. . . . . . . 8 |- (x (_ P~U.x -> (rank` x) (_ (rank` P~U.x))
43	39, 42	ax-mp 7	. . . . . . 7 |- (rank` x) (_ (rank` P~U.x)
44	40	rankpw 4684	. . . . . . 7 |- (rank` P~U.x) = suc (rank` U.x)
45	43, 44	sseqtr 2093	. . . . . 6 |- (rank` x) (_ suc (rank` U.x)
46		uniss 2521	. . . . . 6 |- ((rank` x) (_ suc (rank` U.x) -> U.(rank` x) (_ U.suc (rank` U.x))
47	45, 46	ax-mp 7	. . . . 5 |- U.(rank` x) (_ U.suc (rank` U.x)
48		rankon 4671	. . . . . 6 |- (rank` U.x) e. On
49	48	onunisuc 3106	. . . . 5 |- U.suc (rank` U.x) = (rank` U.x)
50	47, 49	sseqtr 2093	. . . 4 |- U.(rank` x) (_ (rank` U.x)
51	38, 50	eqssi 2078	. . 3 |- (rank` U.x) = U.(rank` x)
52	5, 51	vtoclg 1847	. 2 |- (A e. V -> (rank` U.A) = U.(rank` A))
53		uniexb 2907	. . . . . 6 |- (A e. V <-> U.A e. V)
54	53	negbii 187	. . . . 5 |- (-. A e. V <-> -. U.A e. V)
55		fvprc 3721	. . . . 5 |- (-. U.A e. V -> (rank` U.A) = (/))
56	54, 55	sylbi 199	. . . 4 |- (-. A e. V -> (rank` U.A) = (/))
57		uni0 2525	. . . 4 |- U.(/) = (/)
58	56, 57	syl6eqr 1525	. . 3 |- (-. A e. V -> (rank` U.A) = U.(/))
59		fvprc 3721	. . . 4 |- (-. A e. V -> (rank` A) = (/))
60	59	unieqd 2512	. . 3 |- (-. A e. V -> U.(rank` A) = U.(/))
61	58, 60	eqtr4d 1510	. 2 |- (-. A e. V -> (rank` U.A) = U.(rank` A))
62	52, 61	pm2.61i 126	1 |- (rank` U.A) = U.(rank` A)

Syntax hints:  -. wn 2   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  Vcvv 1811   (_ wss 2047  (/)c0 2280  P~cpw 2401  U.cuni 2503  U_ciun 2566  Oncon0 2948  suc csuc 2950  ` cfv 3182  R1cr1 4641  rankcrnk 4642

This theorem is referenced by:  rankuniss 4701  rankbnd2 4704  rankxplim2 4713  rankxplim3 4714  rankxpsuc 4715

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-r1 4643  df-rank 4644

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