Theorem fvmptnf 5965

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 5966 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 5501	. . . 4 |- dom F C_ D
3	2	sseli 3500	. . 3 |- ( A e. dom F -> A e. D )
4		eqid 2467	. . . . . . 7 |- ( x e. D |-> ( _I ` B ) ) = ( x e. D |-> ( _I ` B ) )
5	1, 4	fvmptex 5958	. . . . . 6 |- ( F ` A ) = ( ( x e. D |-> ( _I ` B ) ) ` A )
6		fvex 5874	. . . . . . 7 |- ( _I ` C ) e. _V
7		fvmptf.1	. . . . . . . 8 |- F/_ x A
8		nfcv 2629	. . . . . . . . 9 |- F/_ x _I
9		fvmptf.2	. . . . . . . . 9 |- F/_ x C
10	8, 9	nffv 5871	. . . . . . . 8 |- F/_ x ( _I ` C )
11		fvmptf.3	. . . . . . . . 9 |- ( x = A -> B = C )
12	11	fveq2d 5868	. . . . . . . 8 |- ( x = A -> ( _I ` B ) = ( _I ` C ) )
13	7, 10, 12, 4	fvmptf 5964	. . . . . . 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 2520	. . . . 5 |- ( A e. D -> ( F ` A ) = ( _I ` C ) )
16		fvprc 5858	. . . . 5 |- ( -. C e. _V -> ( _I ` C ) = (/) )
17	15, 16	sylan9eq 2528	. . . 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 5888	. 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 1379   e. wcel 1767   F/_wnfc 2615   _Vcvv 3113   (/)c0 3785   |-> cmpt 4505   _I cid 4790   dom cdm 4999   ` cfv 5586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594

This theorem is referenced by:  fvmptn  5966  rdgsucmptnf  7092  frsucmptn  7101