Theorem tz7.48-1 2994

Description: Proposition 7.48(1) of [TakeutiZaring] p. 51.

Hypothesis

Ref	Expression
tz7.48.1	|- F Fn On

Assertion

Ref	Expression
tz7.48-1	|- (A.x e. On (F` x) e. (A \ (F"x)) -> ran F (_ A)

Distinct variable group(s):   x,F   x,A

Proof of Theorem tz7.48-1

Step	Hyp	Ref	Expression
1		hbra1 1237	. . . 4 |- (A.x e. On (F` x) e. (A \ (F"x)) -> A.xA.x e. On (F` x) e. (A \ (F"x)))
2		ax-17 925	. . . 4 |- (y e. A -> A.x y e. A)
3		ra4 1243	. . . . 5 |- (A.x e. On (F` x) e. (A \ (F"x)) -> (x e. On -> (F` x) e. (A \ (F"x))))
4		eleq1 1149	. . . . . . . . 9 |- ((F` x) = y -> ((F` x) e. A <-> y e. A))
5		eldifi 1591	. . . . . . . . 9 |- ((F` x) e. (A \ (F"x)) -> (F` x) e. A)
6	4, 5	syl5bi 183	. . . . . . . 8 |- ((F` x) = y -> ((F` x) e. (A \ (F"x)) -> y e. A))
7	6	com12 13	. . . . . . 7 |- ((F` x) e. (A \ (F"x)) -> ((F` x) = y -> y e. A))
8	7	syl3 18	. . . . . 6 |- ((x e. On -> (F` x) e. (A \ (F"x))) -> (x e. On -> ((F` x) = y -> y e. A)))
9	8	imp3a 279	. . . . 5 |- ((x e. On -> (F` x) e. (A \ (F"x))) -> ((x e. On /\ (F` x) = y) -> y e. A))
10	3, 9	syl 12	. . . 4 |- (A.x e. On (F` x) e. (A \ (F"x)) -> ((x e. On /\ (F` x) = y) -> y e. A))
11	1, 2, 10	19.23ad 748	. . 3 |- (A.x e. On (F` x) e. (A \ (F"x)) -> (E.x(x e. On /\ (F` x) = y) -> y e. A))
12		visset 1350	. . . . 5 |- y e. V
13	12	elrn 2562	. . . 4 |- (y e. ran F <-> E.x<.x, y>. e. F)
14		visset 1350	. . . . . . . . 9 |- x e. V
15	14	opeldm 2534	. . . . . . . 8 |- (<.x, y>. e. F -> x e. dom F)
16		tz7.48.1	. . . . . . . . . 10 |- F Fn On
17		fndm 2723	. . . . . . . . . 10 |- (F Fn On -> dom F = On)
18	16, 17	ax-mp 6	. . . . . . . . 9 |- dom F = On
19	18	eleq2i 1153	. . . . . . . 8 |- (x e. dom F <-> x e. On)
20	15, 19	sylib 173	. . . . . . 7 |- (<.x, y>. e. F -> x e. On)
21	20	ancri 245	. . . . . 6 |- (<.x, y>. e. F -> (x e. On /\ <.x, y>. e. F))
22	12	fnfvop 2856	. . . . . . . 8 |- ((F Fn On /\ x e. On) -> ((F` x) = y <-> <.x, y>. e. F))
23	16, 22	mpan 518	. . . . . . 7 |- (x e. On -> ((F` x) = y <-> <.x, y>. e. F))
24	23	pm5.32i 489	. . . . . 6 |- ((x e. On /\ (F` x) = y) <-> (x e. On /\ <.x, y>. e. F))
25	21, 24	sylibr 175	. . . . 5 |- (<.x, y>. e. F -> (x e. On /\ (F` x) = y))
26	25	19.22i 723	. . . 4 |- (E.x<.x, y>. e. F -> E.x(x e. On /\ (F` x) = y))
27	13, 26	sylbi 174	. . 3 |- (y e. ran F -> E.x(x e. On /\ (F` x) = y))
28	11, 27	syl5 22	. 2 |- (A.x e. On (F` x) e. (A \ (F"x)) -> (y e. ran F -> y e. A))
29	28	ssrdv 1509	1 |- (A.x e. On (F` x) e. (A \ (F"x)) -> ran F (_ A)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  A.wral 1201   \ cdif 1484   (_ wss 1487  <.cop 1810  Oncon0 2199  dom cdm 2410  ran crn 2411  "cima 2413   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  tz7.48-3 2996

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  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-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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