Theorem tz7.48-3 3943

Description: Proposition 7.48(3) of [TakeutiZaring] p. 51.

Hypothesis

Ref	Expression
tz7.48.1	|- F Fn On

Assertion

Ref	Expression
tz7.48-3	|- (A.x e. On (F` x) e. (A \ (F"x)) -> -. A e. V)

Distinct variable groups:   x,F   x,A

Proof of Theorem tz7.48-3

Step	Hyp	Ref	Expression
1		onprc 2979	. . . 4 |- -. On e. V
2		tz7.48.1	. . . . . 6 |- F Fn On
3		fndm 3573	. . . . . 6 |- (F Fn On -> dom F = On)
4	2, 3	ax-mp 7	. . . . 5 |- dom F = On
5	4	eleq1i 1529	. . . 4 |- (dom F e. V <-> On e. V)
6	1, 5	mtbir 192	. . 3 |- -. dom F e. V
7	2	tz7.48-2 3942	. . . 4 |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)
8		funrnex 3599	. . . . . 6 |- (dom `' F e. V -> (Fun `'F -> ran `' F e. V))
9	8	com12 11	. . . . 5 |- (Fun `'F -> (dom `' F e. V -> ran `' F e. V))
10		df-rn 3179	. . . . . 6 |- ran F = dom `' F
11	10	eleq1i 1529	. . . . 5 |- (ran F e. V <-> dom `' F e. V)
12		dfdm4 3294	. . . . . 6 |- dom F = ran `' F
13	12	eleq1i 1529	. . . . 5 |- (dom F e. V <-> ran `' F e. V)
14	9, 11, 13	3imtr4g 551	. . . 4 |- (Fun `'F -> (ran F e. V -> dom F e. V))
15	7, 14	syl 10	. . 3 |- (A.x e. On (F` x) e. (A \ (F"x)) -> (ran F e. V -> dom F e. V))
16	6, 15	mtoi 107	. 2 |- (A.x e. On (F` x) e. (A \ (F"x)) -> -. ran F e. V)
17	2	tz7.48-1 3941	. . 3 |- (A.x e. On (F` x) e. (A \ (F"x)) -> ran F (_ A)
18		ssexg 2711	. . . 4 |- ((ran F (_ A /\ A e. V) -> ran F e. V)
19	18	ex 373	. . 3 |- (ran F (_ A -> (A e. V -> ran F e. V))
20	17, 19	syl 10	. 2 |- (A.x e. On (F` x) e. (A \ (F"x)) -> (A e. V -> ran F e. V))
21	16, 20	mtod 108	1 |- (A.x e. On (F` x) e. (A \ (F"x)) -> -. A e. V)

Syntax hints:  -. wn 2   -> wi 3   = wceq 953   e. wcel 955  A.wral 1637  Vcvv 1802   \ cdif 2034   (_ wss 2037  Oncon0 2938  `'ccnv 3159  dom cdm 3160  ran crn 3161  "cima 3163  Fun wfun 3166   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  tz7.49 3944

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-9 962  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-rep 2683  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-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-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fv 3188

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