Theorem fvmptex 6294

Description: Express a function F whose value B may not always be a set in terms of another function G for which sethood is guaranteed. (Note that ( _I ` B ) is just shorthand for if ( B e. _V , B , (/) ), and it is always a set by fvex 6201.) Note also that these functions are not the same; wherever B ( C ) is not a set, C is not in the domain of F (so it evaluates to the empty set), but C is in the domain of G, and G ( C ) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
fvmptex.1	|- F = ( x e. A |-> B )
fvmptex.2	|- G = ( x e. A |-> ( _I ` B ) )

Assertion

Ref	Expression
fvmptex	|- ( F ` C ) = ( G ` C )

Distinct variable group:   x, A

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

Proof of Theorem fvmptex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3536	. . . 4 |- ( y = C -> [_ y / x ]_ B = [_ C / x ]_ B )
2		fvmptex.1	. . . . 5 |- F = ( x e. A |-> B )
3		nfcv 2764	. . . . . 6 |- F/_ y B
4		nfcsb1v 3549	. . . . . 6 |- F/_ x [_ y / x ]_ B
5		csbeq1a 3542	. . . . . 6 |- ( x = y -> B = [_ y / x ]_ B )
6	3, 4, 5	cbvmpt 4749	. . . . 5 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
7	2, 6	eqtri 2644	. . . 4 |- F = ( y e. A |-> [_ y / x ]_ B )
8	1, 7	fvmpti 6281	. . 3 |- ( C e. A -> ( F ` C ) = ( _I ` [_ C / x ]_ B ) )
9	1	fveq2d 6195	. . . 4 |- ( y = C -> ( _I ` [_ y / x ]_ B ) = ( _I ` [_ C / x ]_ B ) )
10		fvmptex.2	. . . . 5 |- G = ( x e. A |-> ( _I ` B ) )
11		nfcv 2764	. . . . . 6 |- F/_ y ( _I ` B )
12		nfcv 2764	. . . . . . 7 |- F/_ x _I
13	12, 4	nffv 6198	. . . . . 6 |- F/_ x ( _I ` [_ y / x ]_ B )
14	5	fveq2d 6195	. . . . . 6 |- ( x = y -> ( _I ` B ) = ( _I ` [_ y / x ]_ B ) )
15	11, 13, 14	cbvmpt 4749	. . . . 5 |- ( x e. A |-> ( _I ` B ) ) = ( y e. A |-> ( _I ` [_ y / x ]_ B ) )
16	10, 15	eqtri 2644	. . . 4 |- G = ( y e. A |-> ( _I ` [_ y / x ]_ B ) )
17		fvex 6201	. . . 4 |- ( _I ` [_ C / x ]_ B ) e. _V
18	9, 16, 17	fvmpt 6282	. . 3 |- ( C e. A -> ( G ` C ) = ( _I ` [_ C / x ]_ B ) )
19	8, 18	eqtr4d 2659	. 2 |- ( C e. A -> ( F ` C ) = ( G ` C ) )
20	2	dmmptss 5631	. . . . . 6 |- dom F C_ A
21	20	sseli 3599	. . . . 5 |- ( C e. dom F -> C e. A )
22	21	con3i 150	. . . 4 |- ( -. C e. A -> -. C e. dom F )
23		ndmfv 6218	. . . 4 |- ( -. C e. dom F -> ( F ` C ) = (/) )
24	22, 23	syl 17	. . 3 |- ( -. C e. A -> ( F ` C ) = (/) )
25		fvex 6201	. . . . . 6 |- ( _I ` B ) e. _V
26	25, 10	dmmpti 6023	. . . . 5 |- dom G = A
27	26	eleq2i 2693	. . . 4 |- ( C e. dom G <-> C e. A )
28		ndmfv 6218	. . . 4 |- ( -. C e. dom G -> ( G ` C ) = (/) )
29	27, 28	sylnbir 321	. . 3 |- ( -. C e. A -> ( G ` C ) = (/) )
30	24, 29	eqtr4d 2659	. 2 |- ( -. C e. A -> ( F ` C ) = ( G ` C ) )
31	19, 30	pm2.61i 176	1 |- ( F ` C ) = ( G ` C )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   [_csb 3533   (/)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:  fvmptnf  6302  sumeq2ii  14423  prodeq2ii  14643