Theorem tz9.12lem1 4631

Description: Lemma for tz9.12 4634.

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 1804	. . . 4 |- y e. V
2	1	elima3 3394	. . 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 1530	. . . . . . 7 |- (<.x, y>. e. F <-> <.x, y>. e. {<.z, w>. | w = |^|{v e. On | z e. (R1` v)}})
5		visset 1804	. . . . . . . 8 |- x e. V
6		eleq1 1526	. . . . . . . . . . 11 |- (z = x -> (z e. (R1` v) <-> x e. (R1` v)))
7	6	rabbisdv 1798	. . . . . . . . . 10 |- (z = x -> {v e. On | z e. (R1` v)} = {v e. On | x e. (R1` v)})
8	7	inteqd 2528	. . . . . . . . 9 |- (z = x -> |^|{v e. On | z e. (R1` v)} = |^|{v e. On | x e. (R1` v)})
9	8	eqeq2d 1478	. . . . . . . 8 |- (z = x -> (w = |^|{v e. On | z e. (R1` v)} <-> w = |^|{v e. On | x e. (R1` v)}))
10		eqeq1 1473	. . . . . . . 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 2809	. . . . . . 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 1025	. . . . . . . 8 |- (y = |^|{v e. On | x e. (R1` v)} -> E.y y = |^|{v e. On | x e. (R1` v)})
14		isset 1805	. . . . . . . 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 2719	. . . . . . . 8 |- ({v e. On | x e. (R1` v)} =/= (/) <-> |^|{v e. On | x e. (R1` v)} e. V)
17		eleq1 1526	. . . . . . . . 9 |- (y = |^|{v e. On | x e. (R1` v)} -> (y e. On <-> |^|{v e. On | x e. (R1` v)} e. On))
18		ssrab2 2121	. . . . . . . . . 10 |- {v e. On | x e. (R1` v)} (_ On
19		oninton 3002	. . . . . . . . . 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 693	. . . . . . . . 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 1290	. . 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 2059	1 |- (F"A) (_ On

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977   =/= wne 1577  {crab 1640  Vcvv 1802   (_ wss 2037  (/)c0 2270  <.cop 2401  |^|cint 2523  {copab 2656  Oncon0 2938  "cima 3163  ` cfv 3172  R1cr1 4613

This theorem is referenced by:  tz9.12lem2 4632  tz9.12lem3 4633

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181

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