Theorem tz9.12lem1 3503

Description: Lemma for tz9.12 3506.

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(s):   z,w,v,A

Proof of Theorem tz9.12lem1

Step	Hyp	Ref	Expression
1		visset 1350	. . . 4 |- y e. V
2	1	elima3 2608	. . 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 1153	. . . . . . 7 |- (<.x, y>. e. F <-> <.x, y>. e. {<.z, w>. | w = |^|{v e. On | z e. (R1` v)}})
5		visset 1350	. . . . . . . 8 |- x e. V
6		eleq1 1149	. . . . . . . . . . 11 |- (z = x -> (z e. (R1` v) <-> x e. (R1` v)))
7	6	birabsdv 1344	. . . . . . . . . 10 |- (z = x -> {v e. On | z e. (R1` v)} = {v e. On | x e. (R1` v)})
8	7	inteqd 1970	. . . . . . . . 9 |- (z = x -> |^|{v e. On | z e. (R1` v)} = |^|{v e. On | x e. (R1` v)})
9	8	cleq2d 1112	. . . . . . . 8 |- (z = x -> (w = |^|{v e. On | z e. (R1` v)} <-> w = |^|{v e. On | x e. (R1` v)}))
10		cleq1 1107	. . . . . . . 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 2117	. . . . . . 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 151	. . . . . 6 |- (<.x, y>. e. F <-> y = |^|{v e. On | x e. (R1` v)})
13		19.8a 712	. . . . . . . 8 |- (y = |^|{v e. On | x e. (R1` v)} -> E.y y = |^|{v e. On | x e. (R1` v)})
14		isset 1351	. . . . . . . 8 |- (|^|{v e. On | x e. (R1` v)} e. V <-> E.y y = |^|{v e. On | x e. (R1` v)})
15	13, 14	sylibr 175	. . . . . . 7 |- (y = |^|{v e. On | x e. (R1` v)} -> |^|{v e. On | x e. (R1` v)} e. V)
16		intex 1986	. . . . . . . 8 |- (-. {v e. On | x e. (R1` v)} = (/) <-> |^|{v e. On | x e. (R1` v)} e. V)
17		eleq1 1149	. . . . . . . . . 10 |- (y = |^|{v e. On | x e. (R1` v)} -> (y e. On <-> |^|{v e. On | x e. (R1` v)} e. On))
18		ssrab 1556	. . . . . . . . . . 11 |- {v e. On | x e. (R1` v)} (_ On
19		oninton 2267	. . . . . . . . . . 11 |- (({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 518	. . . . . . . . . 10 |- (-. {v e. On | x e. (R1` v)} = (/) -> |^|{v e. On | x e. (R1` v)} e. On)
21	17, 20	syl5bir 184	. . . . . . . . 9 |- (y = |^|{v e. On | x e. (R1` v)} -> (-. {v e. On | x e. (R1` v)} = (/) -> y e. On))
22	21	com12 13	. . . . . . . 8 |- (-. {v e. On | x e. (R1` v)} = (/) -> (y = |^|{v e. On | x e. (R1` v)} -> y e. On))
23	16, 22	sylbir 176	. . . . . . 7 |- (|^|{v e. On | x e. (R1` v)} e. V -> (y = |^|{v e. On | x e. (R1` v)} -> y e. On))
24	15, 23	mpcom 49	. . . . . 6 |- (y = |^|{v e. On | x e. (R1` v)} -> y e. On)
25	12, 24	sylbi 174	. . . . 5 |- (<.x, y>. e. F -> y e. On)
26	25	adantl 305	. . . 4 |- ((x e. A /\ <.x, y>. e. F) -> y e. On)
27	26	19.23aiv 952	. . 3 |- (E.x(x e. A /\ <.x, y>. e. F) -> y e. On)
28	2, 27	sylbi 174	. 2 |- (y e. (F"A) -> y e. On)
29	28	ssriv 1508	1 |- (F"A) (_ On

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  E.wex 678   = weq 797   = wceq 1091   e. wcel 1092  {crab 1204  Vcvv 1348   (_ wss 1487  (/)c0 1707  <.cop 1810  |^|cint 1965  {copab 2055  Oncon0 2199  "cima 2413  ` cfv 2422  R1cr1 3485

This theorem is referenced by:  tz9.12lem2 3504  tz9.12lem3 3505

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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