Theorem fvmptnf 5982

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 5983 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 5338	. . . 4 |- dom F C_ D
3	2	sseli 3414	. . 3 |- ( A e. dom F -> A e. D )
4		eqid 2471	. . . . . . 7 |- ( x e. D |-> ( _I ` B ) ) = ( x e. D |-> ( _I ` B ) )
5	1, 4	fvmptex 5975	. . . . . 6 |- ( F ` A ) = ( ( x e. D |-> ( _I ` B ) ) ` A )
6		fvex 5889	. . . . . . 7 |- ( _I ` C ) e. _V
7		fvmptf.1	. . . . . . . 8 |- F/_ x A
8		nfcv 2612	. . . . . . . . 9 |- F/_ x _I
9		fvmptf.2	. . . . . . . . 9 |- F/_ x C
10	8, 9	nffv 5886	. . . . . . . 8 |- F/_ x ( _I ` C )
11		fvmptf.3	. . . . . . . . 9 |- ( x = A -> B = C )
12	11	fveq2d 5883	. . . . . . . 8 |- ( x = A -> ( _I ` B ) = ( _I ` C ) )
13	7, 10, 12, 4	fvmptf 5981	. . . . . . 7 |- ( ( A e. D /\ ( _I ` C ) e. _V ) -> ( ( x e. D |-> ( _I ` B ) ) ` A ) = ( _I ` C ) )
14	6, 13	mpan2 685	. . . . . 6 |- ( A e. D -> ( ( x e. D |-> ( _I ` B ) ) ` A ) = ( _I ` C ) )
15	5, 14	syl5eq 2517	. . . . 5 |- ( A e. D -> ( F ` A ) = ( _I ` C ) )
16		fvprc 5873	. . . . 5 |- ( -. C e. _V -> ( _I ` C ) = (/) )
17	15, 16	sylan9eq 2525	. . . 4 |- ( ( A e. D /\ -. C e. _V ) -> ( F ` A ) = (/) )
18	17	expcom 442	. . 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 5903	. 2 |- ( -. A e. dom F -> ( F ` A ) = (/) )
21	19, 20	pm2.61d1 164	1 |- ( -. C e. _V -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1452   e. wcel 1904   F/_wnfc 2599   _Vcvv 3031   (/)c0 3722   |-> cmpt 4454   _I cid 4749   dom cdm 4839   ` cfv 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597

This theorem is referenced by:  fvmptn  5983  rdgsucmptnf  7165  frsucmptn  7174