Theorem tz9.12lem1 4659

Description: Lemma for tz9.12 4662.

Hypotheses

Ref	Expression
tz9.12lem.1	|- A e. V
tz9.12lem.2	|- F = {<.z, w>. | w = |^|{v e. On | z e. (R1` v)}}

Assertion

Ref	Expression
tz9.12lem1	|- (F"A) (_ On

Distinct variable group:   z,w,v,A

Proof of Theorem tz9.12lem1

Step	Hyp	Ref	Expression
1		visset 1813	. . . 4 |- y e. V
2	1	elima3 3410	. . 3 |- (y e. (F"A) <-> E.x(x e. A /\ <.x, y>. e. F))
3		tz9.12lem.2	. . . . . . . 8 |- F = {<.z, w>. | w = |^|{v e. On | z e. (R1` v)}}
4	3	eleq2i 1538	. . . . . . 7 |- (<.x, y>. e. F <-> <.x, y>. e. {<.z, w>. | w = |^|{v e. On | z e. (R1` v)}})
5		visset 1813	. . . . . . . 8 |- x e. V
6		eleq1 1534	. . . . . . . . . . 11 |- (z = x -> (z e. (R1` v) <-> x e. (R1` v)))
7	6	rabbisdv 1807	. . . . . . . . . 10 |- (z = x -> {v e. On | z e. (R1` v)} = {v e. On | x e. (R1` v)})
8	7	inteqd 2538	. . . . . . . . 9 |- (z = x -> |^|{v e. On | z e. (R1` v)} = |^|{v e. On | x e. (R1` v)})
9	8	eqeq2d 1486	. . . . . . . 8 |- (z = x -> (w = |^|{v e. On | z e. (R1` v)} <-> w = |^|{v e. On | x e. (R1` v)}))
10		eqeq1 1481	. . . . . . . 8 |- (w = y -> (w = |^|{v e. On | x e. (R1` v)} <-> y = |^|{v e. On | x e. (R1` v)}))
11	5, 1, 9, 10	opelopab 2820	. . . . . . 7 |- (<.x, y>. e. {<.z, w>. | w = |^|{v e. On | z e. (R1` v)}} <-> y = |^|{v e. On | x e. (R1` v)})
12	4, 11	bitr 173	. . . . . 6 |- (<.x, y>. e. F <-> y = |^|{v e. On | x e. (R1` v)})
13		19.8a 1029	. . . . . . . 8 |- (y = |^|{v e. On | x e. (R1` v)} -> E.y y = |^|{v e. On | x e. (R1` v)})
14		isset 1814	. . . . . . . 8 |- (|^|{v e. On | x e. (R1` v)} e. V <-> E.y y = |^|{v e. On | x e. (R1` v)})
15	13, 14	sylibr 200	. . . . . . 7 |- (y = |^|{v e. On | x e. (R1` v)} -> |^|{v e. On | x e. (R1` v)} e. V)
16		intex 2729	. . . . . . . 8 |- ({v e. On | x e. (R1` v)} =/= (/) <-> |^|{v e. On | x e. (R1` v)} e. V)
17		eleq1 1534	. . . . . . . . 9 |- (y = |^|{v e. On | x e. (R1` v)} -> (y e. On <-> |^|{v e. On | x e. (R1` v)} e. On))
18		ssrab2 2131	. . . . . . . . . 10 |- {v e. On | x e. (R1` v)} (_ On
19		oninton 3012	. . . . . . . . . 10 |- (({v e. On | x e. (R1` v)} (_ On /\ {v e. On | x e. (R1` v)} =/= (/)) -> |^|{v e. On | x e. (R1` v)} e. On)
20	18, 19	mpan 695	. . . . . . . . 9 |- ({v e. On | x e. (R1` v)} =/= (/) -> |^|{v e. On | x e. (R1` v)} e. On)
21	17, 20	syl5cbir 211	. . . . . . . 8 |- ({v e. On | x e. (R1` v)} =/= (/) -> (y = |^|{v e. On | x e. (R1` v)} -> y e. On))
22	16, 21	sylbir 201	. . . . . . 7 |- (|^|{v e. On | x e. (R1` v)} e. V -> (y = |^|{v e. On | x e. (R1` v)} -> y e. On))
23	15, 22	mpcom 49	. . . . . 6 |- (y = |^|{v e. On | x e. (R1` v)} -> y e. On)
24	12, 23	sylbi 199	. . . . 5 |- (<.x, y>. e. F -> y e. On)
25	24	adantl 388	. . . 4 |- ((x e. A /\ <.x, y>. e. F) -> y e. On)
26	25	19.23aiv 1295	. . 3 |- (E.x(x e. A /\ <.x, y>. e. F) -> y e. On)
27	2, 26	sylbi 199	. 2 |- (y e. (F"A) -> y e. On)
28	27	ssriv 2069	1 |- (F"A) (_ On

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980   =/= wne 1585  {crab 1648  Vcvv 1811   (_ wss 2047  (/)c0 2280  <.cop 2411  |^|cint 2533  {copab 2666  Oncon0 2948  "cima 3173  ` cfv 3182  R1cr1 4641

This theorem is referenced by:  tz9.12lem2 4660  tz9.12lem3 4661

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-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  ax-un 2866

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-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

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