Theorem foeq123d 6132

Description: Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
f1eq123d.1	|- ( ph -> F = G )
f1eq123d.2	|- ( ph -> A = B )
f1eq123d.3	|- ( ph -> C = D )

Assertion

Ref	Expression
foeq123d	|- ( ph -> ( F : A -onto-> C <-> G : B -onto-> D ) )

Proof of Theorem foeq123d

Step	Hyp	Ref	Expression
1		f1eq123d.1	. . 3 |- ( ph -> F = G )
2		foeq1 6111	. . 3 |- ( F = G -> ( F : A -onto-> C <-> G : A -onto-> C ) )
3	1, 2	syl 17	. 2 |- ( ph -> ( F : A -onto-> C <-> G : A -onto-> C ) )
4		f1eq123d.2	. . 3 |- ( ph -> A = B )
5		foeq2 6112	. . 3 |- ( A = B -> ( G : A -onto-> C <-> G : B -onto-> C ) )
6	4, 5	syl 17	. 2 |- ( ph -> ( G : A -onto-> C <-> G : B -onto-> C ) )
7		f1eq123d.3	. . 3 |- ( ph -> C = D )
8		foeq3 6113	. . 3 |- ( C = D -> ( G : B -onto-> C <-> G : B -onto-> D ) )
9	7, 8	syl 17	. 2 |- ( ph -> ( G : B -onto-> C <-> G : B -onto-> D ) )
10	3, 6, 9	3bitrd 294	1 |- ( ph -> ( F : A -onto-> C <-> G : B -onto-> D ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   -onto->wfo 5886

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-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-fo 5894

This theorem is referenced by:  fullfo  16572  cofull  16594  resgrpplusfrn  17436  efabl  24296  iseupth  27061