Theorem fvmptnf 5974

Description: The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 5975 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
fvmptf.1	|- F/_ x A
fvmptf.2	|- F/_ x C
fvmptf.3	|- ( x = A -> B = C )
fvmptf.4	|- F = ( x e. D |-> B )

Assertion

Ref	Expression
fvmptnf	|- ( -. C e. _V -> ( F ` A ) = (/) )

Distinct variable group:   x, D

Allowed substitution hints:   A( x)   B( x)   C( x)   F( x)

Proof of Theorem fvmptnf

Step	Hyp	Ref	Expression
1		fvmptf.4	. . . . 5 |- F = ( x e. D |-> B )
2	1	dmmptss 5509	. . . 4 |- dom F C_ D
3	2	sseli 3495	. . 3 |- ( A e. dom F -> A e. D )
4		eqid 2457	. . . . . . 7 |- ( x e. D |-> ( _I ` B ) ) = ( x e. D |-> ( _I ` B ) )
5	1, 4	fvmptex 5967	. . . . . 6 |- ( F ` A ) = ( ( x e. D |-> ( _I ` B ) ) ` A )
6		fvex 5882	. . . . . . 7 |- ( _I ` C ) e. _V
7		fvmptf.1	. . . . . . . 8 |- F/_ x A
8		nfcv 2619	. . . . . . . . 9 |- F/_ x _I
9		fvmptf.2	. . . . . . . . 9 |- F/_ x C
10	8, 9	nffv 5879	. . . . . . . 8 |- F/_ x ( _I ` C )
11		fvmptf.3	. . . . . . . . 9 |- ( x = A -> B = C )
12	11	fveq2d 5876	. . . . . . . 8 |- ( x = A -> ( _I ` B ) = ( _I ` C ) )
13	7, 10, 12, 4	fvmptf 5973	. . . . . . 7 |- ( ( A e. D /\ ( _I ` C ) e. _V ) -> ( ( x e. D |-> ( _I ` B ) ) ` A ) = ( _I ` C ) )
14	6, 13	mpan2 671	. . . . . 6 |- ( A e. D -> ( ( x e. D |-> ( _I ` B ) ) ` A ) = ( _I ` C ) )
15	5, 14	syl5eq 2510	. . . . 5 |- ( A e. D -> ( F ` A ) = ( _I ` C ) )
16		fvprc 5866	. . . . 5 |- ( -. C e. _V -> ( _I ` C ) = (/) )
17	15, 16	sylan9eq 2518	. . . 4 |- ( ( A e. D /\ -. C e. _V ) -> ( F ` A ) = (/) )
18	17	expcom 435	. . 3 |- ( -. C e. _V -> ( A e. D -> ( F ` A ) = (/) ) )
19	3, 18	syl5 32	. 2 |- ( -. C e. _V -> ( A e. dom F -> ( F ` A ) = (/) ) )
20		ndmfv 5896	. 2 |- ( -. A e. dom F -> ( F ` A ) = (/) )
21	19, 20	pm2.61d1 159	1 |- ( -. C e. _V -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   e. wcel 1819   F/_wnfc 2605   _Vcvv 3109   (/)c0 3793   |-> cmpt 4515   _I cid 4799   dom cdm 5008   ` cfv 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602

This theorem is referenced by:  fvmptn  5975  rdgsucmptnf  7113  frsucmptn  7122