Theorem tz7.48-3 7539

Description: Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)

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 6984	. . . 4 |- -. On e. _V
2		tz7.48.1	. . . . . 6 |- F Fn On
3		fndm 5990	. . . . . 6 |- ( F Fn On -> dom F = On )
4	2, 3	ax-mp 5	. . . . 5 |- dom F = On
5	4	eleq1i 2692	. . . 4 |- ( dom F e. _V <-> On e. _V )
6	1, 5	mtbir 313	. . 3 |- -. dom F e. _V
7	2	tz7.48-2 7537	. . . 4 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> Fun `' F )
8		funrnex 7133	. . . . . 6 |- ( dom `' F e. _V -> ( Fun `' F -> ran `' F e. _V ) )
9	8	com12 32	. . . . 5 |- ( Fun `' F -> ( dom `' F e. _V -> ran `' F e. _V ) )
10		df-rn 5125	. . . . . 6 |- ran F = dom `' F
11	10	eleq1i 2692	. . . . 5 |- ( ran F e. _V <-> dom `' F e. _V )
12		dfdm4 5316	. . . . . 6 |- dom F = ran `' F
13	12	eleq1i 2692	. . . . 5 |- ( dom F e. _V <-> ran `' F e. _V )
14	9, 11, 13	3imtr4g 285	. . . 4 |- ( Fun `' F -> ( ran F e. _V -> dom F e. _V ) )
15	7, 14	syl 17	. . 3 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> ( ran F e. _V -> dom F e. _V ) )
16	6, 15	mtoi 190	. 2 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> -. ran F e. _V )
17	2	tz7.48-1 7538	. . 3 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> ran F C_ A )
18		ssexg 4804	. . . 4 |- ( ( ran F C_ A /\ A e. _V ) -> ran F e. _V )
19	18	ex 450	. . 3 |- ( ran F C_ A -> ( A e. _V -> ran F e. _V ) )
20	17, 19	syl 17	. 2 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> ( A e. _V -> ran F e. _V ) )
21	16, 20	mtod 189	1 |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> -. A e. _V )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  tz7.49  7540