Theorem fvopab4gf 3781

Description: Value of a function given by an ordered-pair class abstraction. This version of fvopab4g 3779 uses bound-variable hypotheses instead of distinct variable conditions.

Hypotheses

Ref	Expression
fvopab4gf.1	|- (z e. A -> A.x z e. A)
fvopab4gf.2	|- (z e. C -> A.x z e. C)
fvopab4gf.3	|- (x = A -> B = C)
fvopab4gf.4	|- F = {<.x, y>. | (x e. D /\ y = B)}

Assertion

Ref	Expression
fvopab4gf	|- ((A e. D /\ C e. R) -> (F` A) = C)

Distinct variable groups:   z,A   y,B   z,C   x,y,D   x,z

Proof of Theorem fvopab4gf

Step	Hyp	Ref	Expression
1		ax-17 971	. . . . 5 |- (z e. w -> A.x z e. w)
2		fvopab4gf.1	. . . . 5 |- (z e. A -> A.x z e. A)
3		visset 1813	. . . . 5 |- w e. V
4	1, 2, 3	eqvincf 1884	. . . 4 |- (w = A <-> E.x(x = w /\ x = A))
5		ax-17 971	. . . . . . 7 |- (v e. w -> A.x v e. w)
6	3, 5	hbcsb1 2025	. . . . . 6 |- (v e. [_w / x]_B -> A.x v e. [_w / x]_B)
7		fvopab4gf.2	. . . . . 6 |- (z e. C -> A.x z e. C)
8	6, 7	hbeq 1565	. . . . 5 |- ([_w / x]_B = C -> A.x[_w / x]_B = C)
9		csbeq1a 2006	. . . . . 6 |- (x = w -> B = [_w / x]_B)
10		fvopab4gf.3	. . . . . 6 |- (x = A -> B = C)
11	9, 10	sylan9req 1528	. . . . 5 |- ((x = w /\ x = A) -> [_w / x]_B = C)
12	8, 11	19.23ai 1064	. . . 4 |- (E.x(x = w /\ x = A) -> [_w / x]_B = C)
13	4, 12	sylbi 199	. . 3 |- (w = A -> [_w / x]_B = C)
14		eqid 1475	. . 3 |- {<.w, v>. | (w e. D /\ v = [_w / x]_B)} = {<.w, v>. | (w e. D /\ v = [_w / x]_B)}
15	13, 14	fvopab4g 3779	. 2 |- ((A e. D /\ C e. R) -> ({<.w, v>. | (w e. D /\ v = [_w / x]_B)}` A) = C)
16		fvopab4gf.4	. . . 4 |- F = {<.x, y>. | (x e. D /\ y = B)}
17	16	fveq1i 3725	. . 3 |- (F` A) = ({<.x, y>. | (x e. D /\ y = B)}` A)
18		ax-17 971	. . . . 5 |- ((x e. D /\ y = B) -> A.w(x e. D /\ y = B))
19		ax-17 971	. . . . 5 |- ((x e. D /\ y = B) -> A.v(x e. D /\ y = B))
20		ax-17 971	. . . . . 6 |- (w e. D -> A.x w e. D)
21	6	hbeleq 1567	. . . . . 6 |- (v = [_w / x]_B -> A.x v = [_w / x]_B)
22	20, 21	hban 1009	. . . . 5 |- ((w e. D /\ v = [_w / x]_B) -> A.x(w e. D /\ v = [_w / x]_B))
23		ax-17 971	. . . . 5 |- ((w e. D /\ v = [_w / x]_B) -> A.y(w e. D /\ v = [_w / x]_B))
24		eleq1 1534	. . . . . . 7 |- (x = w -> (x e. D <-> w e. D))
25	24	adantr 389	. . . . . 6 |- ((x = w /\ y = v) -> (x e. D <-> w e. D))
26		id 59	. . . . . . 7 |- (y = v -> y = v)
27	26, 9	eqeqan12rd 1491	. . . . . 6 |- ((x = w /\ y = v) -> (y = B <-> v = [_w / x]_B))
28	25, 27	anbi12d 628	. . . . 5 |- ((x = w /\ y = v) -> ((x e. D /\ y = B) <-> (w e. D /\ v = [_w / x]_B)))
29	18, 19, 22, 23, 28	cbvopab 2672	. . . 4 |- {<.x, y>. | (x e. D /\ y = B)} = {<.w, v>. | (w e. D /\ v = [_w / x]_B)}
30	29	fveq1i 3725	. . 3 |- ({<.x, y>. | (x e. D /\ y = B)}` A) = ({<.w, v>. | (w e. D /\ v = [_w / x]_B)}` A)
31	17, 30	eqtr 1495	. 2 |- (F` A) = ({<.w, v>. | (w e. D /\ v = [_w / x]_B)}` A)
32	15, 31	syl5eq 1519	1 |- ((A e. D /\ C e. R) -> (F` A) = C)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  [_csb 2001  {copab 2666  ` cfv 3182

This theorem is referenced by:  fvopab4sf 3782

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-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  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-fv 3198

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