Theorem tz7.48-3 5167

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 3865	. . . 4 |- -. On e. _V
2		tz7.48.1	. . . . . 6 |- F Fn On
3		fndm 4512	. . . . . 6 |- (F Fn On -> dom F = On)
4	2, 3	ax-mp 7	. . . . 5 |- dom F = On
5	4	eleq1i 1960	. . . 4 |- (dom F e. _V <-> On e. _V)
6	1, 5	mtbir 209	. . 3 |- -. dom F e. _V
7	2	tz7.48-2 5166	. . . 4 |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)
8		funrnex 4544	. . . . . 6 |- (dom `' F e. _V -> (Fun `'F -> ran `' F e. _V))
9	8	com12 14	. . . . 5 |- (Fun `'F -> (dom `' F e. _V -> ran `' F e. _V))
10		df-rn 4005	. . . . . 6 |- ran F = dom `' F
11	10	eleq1i 1960	. . . . 5 |- (ran F e. _V <-> dom `' F e. _V)
12		dfdm4 4151	. . . . . 6 |- dom F = ran `' F
13	12	eleq1i 1960	. . . . 5 |- (dom F e. _V <-> ran `' F e. _V)
14	9, 11, 13	3imtr4g 612	. . . 4 |- (Fun `'F -> (ran F e. _V -> dom F e. _V))
15	7, 14	syl 12	. . 3 |- (A.x e. On (F` x) e. (A \ (F"x)) -> (ran F e. _V -> dom F e. _V))
16	6, 15	mtoi 122	. 2 |- (A.x e. On (F` x) e. (A \ (F"x)) -> -. ran F e. _V)
17	2	tz7.48-1 5165	. . 3 |- (A.x e. On (F` x) e. (A \ (F"x)) -> ran F C_ A)
18		ssexg 3457	. . . 4 |- ((ran F C_ A /\ A e. _V) -> ran F e. _V)
19	18	ex 402	. . 3 |- (ran F C_ A -> (A e. _V -> ran F e. _V))
20	17, 19	syl 12	. 2 |- (A.x e. On (F` x) e. (A \ (F"x)) -> (A e. _V -> ran F e. _V))
21	16, 20	mtod 123	1 |- (A.x e. On (F` x) e. (A \ (F"x)) -> -. A e. _V)

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   \ cdif 2590   C_ wss 2593  Oncon0 3657  `'ccnv 3985  dom cdm 3986  ran crn 3987  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  tz7.49 5168

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fv 4014

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