Theorem dmmptdf 39417

Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
dmmptdf.x	|- F/ x ph
dmmptdf.a	|- A = ( x e. B |-> C )
dmmptdf.c	|- ( ( ph /\ x e. B ) -> C e. V )

Assertion

Ref	Expression
dmmptdf	|- ( ph -> dom A = B )

Distinct variable group:   x, B

Allowed substitution hints:   ph( x)   A( x)   C( x)   V( x)

Proof of Theorem dmmptdf

Step	Hyp	Ref	Expression
1		dmmptdf.x	. . . 4 |- F/ x ph
2		dmmptdf.c	. . . . . 6 |- ( ( ph /\ x e. B ) -> C e. V )
3		elex 3212	. . . . . 6 |- ( C e. V -> C e. _V )
4	2, 3	syl 17	. . . . 5 |- ( ( ph /\ x e. B ) -> C e. _V )
5	4	ex 450	. . . 4 |- ( ph -> ( x e. B -> C e. _V ) )
6	1, 5	ralrimi 2957	. . 3 |- ( ph -> A. x e. B C e. _V )
7		rabid2 3118	. . 3 |- ( B = { x e. B | C e. _V } <-> A. x e. B C e. _V )
8	6, 7	sylibr 224	. 2 |- ( ph -> B = { x e. B | C e. _V } )
9		dmmptdf.a	. . 3 |- A = ( x e. B |-> C )
10	9	dmmpt 5630	. 2 |- dom A = { x e. B | C e. _V }
11	8, 10	syl6reqr 2675	1 |- ( ph -> dom A = B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   |-> cmpt 4729   dom cdm 5114

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  smfpimltmpt  40955  smfpimltxrmpt  40967  smfadd  40973  smfpimgtmpt  40989  smfpimgtxrmpt  40992  smfpimioompt  40993  smfrec  40996  smfmul  41002  smfmulc1  41003  smffmpt  41011  smfsupmpt  41021  smfinfmpt  41025  smflimsupmpt  41035  smfliminfmpt  41038