Theorem fvmptnf 6302

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 6303 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 5631	. . . 4 |- dom F C_ D
3	2	sseli 3599	. . 3 |- ( A e. dom F -> A e. D )
4		eqid 2622	. . . . . . 7 |- ( x e. D |-> ( _I ` B ) ) = ( x e. D |-> ( _I ` B ) )
5	1, 4	fvmptex 6294	. . . . . 6 |- ( F ` A ) = ( ( x e. D |-> ( _I ` B ) ) ` A )
6		fvex 6201	. . . . . . 7 |- ( _I ` C ) e. _V
7		fvmptf.1	. . . . . . . 8 |- F/_ x A
8		nfcv 2764	. . . . . . . . 9 |- F/_ x _I
9		fvmptf.2	. . . . . . . . 9 |- F/_ x C
10	8, 9	nffv 6198	. . . . . . . 8 |- F/_ x ( _I ` C )
11		fvmptf.3	. . . . . . . . 9 |- ( x = A -> B = C )
12	11	fveq2d 6195	. . . . . . . 8 |- ( x = A -> ( _I ` B ) = ( _I ` C ) )
13	7, 10, 12, 4	fvmptf 6301	. . . . . . 7 |- ( ( A e. D /\ ( _I ` C ) e. _V ) -> ( ( x e. D |-> ( _I ` B ) ) ` A ) = ( _I ` C ) )
14	6, 13	mpan2 707	. . . . . 6 |- ( A e. D -> ( ( x e. D |-> ( _I ` B ) ) ` A ) = ( _I ` C ) )
15	5, 14	syl5eq 2668	. . . . 5 |- ( A e. D -> ( F ` A ) = ( _I ` C ) )
16		fvprc 6185	. . . . 5 |- ( -. C e. _V -> ( _I ` C ) = (/) )
17	15, 16	sylan9eq 2676	. . . 4 |- ( ( A e. D /\ -. C e. _V ) -> ( F ` A ) = (/) )
18	17	expcom 451	. . 3 |- ( -. C e. _V -> ( A e. D -> ( F ` A ) = (/) ) )
19	3, 18	syl5 34	. 2 |- ( -. C e. _V -> ( A e. dom F -> ( F ` A ) = (/) ) )
20		ndmfv 6218	. 2 |- ( -. A e. dom F -> ( F ` A ) = (/) )
21	19, 20	pm2.61d1 171	1 |- ( -. C e. _V -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   _I cid 5023   dom cdm 5114   ` 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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  fvmptn  6303  rdgsucmptnf  7525  frsucmptn  7534