Theorem fvmptnf 5967

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 5968 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 5331	. . . 4 |- dom F C_ D
3	2	sseli 3428	. . 3 |- ( A e. dom F -> A e. D )
4		eqid 2451	. . . . . . 7 |- ( x e. D |-> ( _I ` B ) ) = ( x e. D |-> ( _I ` B ) )
5	1, 4	fvmptex 5960	. . . . . 6 |- ( F ` A ) = ( ( x e. D |-> ( _I ` B ) ) ` A )
6		fvex 5875	. . . . . . 7 |- ( _I ` C ) e. _V
7		fvmptf.1	. . . . . . . 8 |- F/_ x A
8		nfcv 2592	. . . . . . . . 9 |- F/_ x _I
9		fvmptf.2	. . . . . . . . 9 |- F/_ x C
10	8, 9	nffv 5872	. . . . . . . 8 |- F/_ x ( _I ` C )
11		fvmptf.3	. . . . . . . . 9 |- ( x = A -> B = C )
12	11	fveq2d 5869	. . . . . . . 8 |- ( x = A -> ( _I ` B ) = ( _I ` C ) )
13	7, 10, 12, 4	fvmptf 5966	. . . . . . 7 |- ( ( A e. D /\ ( _I ` C ) e. _V ) -> ( ( x e. D |-> ( _I ` B ) ) ` A ) = ( _I ` C ) )
14	6, 13	mpan2 677	. . . . . 6 |- ( A e. D -> ( ( x e. D |-> ( _I ` B ) ) ` A ) = ( _I ` C ) )
15	5, 14	syl5eq 2497	. . . . 5 |- ( A e. D -> ( F ` A ) = ( _I ` C ) )
16		fvprc 5859	. . . . 5 |- ( -. C e. _V -> ( _I ` C ) = (/) )
17	15, 16	sylan9eq 2505	. . . 4 |- ( ( A e. D /\ -. C e. _V ) -> ( F ` A ) = (/) )
18	17	expcom 437	. . 3 |- ( -. C e. _V -> ( A e. D -> ( F ` A ) = (/) ) )
19	3, 18	syl5 33	. 2 |- ( -. C e. _V -> ( A e. dom F -> ( F ` A ) = (/) ) )
20		ndmfv 5889	. 2 |- ( -. A e. dom F -> ( F ` A ) = (/) )
21	19, 20	pm2.61d1 163	1 |- ( -. C e. _V -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1444   e. wcel 1887   F/_wnfc 2579   _Vcvv 3045   (/)c0 3731   |-> cmpt 4461   _I cid 4744   dom cdm 4834   ` cfv 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590

This theorem is referenced by:  fvmptn  5968  rdgsucmptnf  7147  frsucmptn  7156