Theorem dfateq12d 41209

Description: Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypotheses

Ref	Expression
dfateq12d.1	|- ( ph -> F = G )
dfateq12d.2	|- ( ph -> A = B )

Assertion

Ref	Expression
dfateq12d	|- ( ph -> ( F defAt A <-> G defAt B ) )

Proof of Theorem dfateq12d

Step	Hyp	Ref	Expression
1		dfateq12d.2	. . . 4 |- ( ph -> A = B )
2		dfateq12d.1	. . . . 5 |- ( ph -> F = G )
3	2	dmeqd 5326	. . . 4 |- ( ph -> dom F = dom G )
4	1, 3	eleq12d 2695	. . 3 |- ( ph -> ( A e. dom F <-> B e. dom G ) )
5	1	sneqd 4189	. . . . 5 |- ( ph -> { A } = { B } )
6	2, 5	reseq12d 5397	. . . 4 |- ( ph -> ( F |` { A } ) = ( G |` { B } ) )
7	6	funeqd 5910	. . 3 |- ( ph -> ( Fun ( F |` { A } ) <-> Fun ( G |` { B } ) ) )
8	4, 7	anbi12d 747	. 2 |- ( ph -> ( ( A e. dom F /\ Fun ( F |` { A } ) ) <-> ( B e. dom G /\ Fun ( G |` { B } ) ) ) )
9		df-dfat 41196	. 2 |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
10		df-dfat 41196	. 2 |- ( G defAt B <-> ( B e. dom G /\ Fun ( G |` { B } ) ) )
11	8, 9, 10	3bitr4g 303	1 |- ( ph -> ( F defAt A <-> G defAt B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   dom cdm 5114   |` cres 5116   Fun wfun 5882   defAt wdfat 41193

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-fun 5890  df-dfat 41196

This theorem is referenced by:  afveq12d  41213