Theorem f1eq123d 6131

Description: Equality deduction for one-to-one 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
f1eq123d	|- ( ph -> ( F : A -1-1-> C <-> G : B -1-1-> D ) )

Proof of Theorem f1eq123d

Step	Hyp	Ref	Expression
1		f1eq123d.1	. . 3 |- ( ph -> F = G )
2		f1eq1 6096	. . 3 |- ( F = G -> ( F : A -1-1-> C <-> G : A -1-1-> C ) )
3	1, 2	syl 17	. 2 |- ( ph -> ( F : A -1-1-> C <-> G : A -1-1-> C ) )
4		f1eq123d.2	. . 3 |- ( ph -> A = B )
5		f1eq2 6097	. . 3 |- ( A = B -> ( G : A -1-1-> C <-> G : B -1-1-> C ) )
6	4, 5	syl 17	. 2 |- ( ph -> ( G : A -1-1-> C <-> G : B -1-1-> C ) )
7		f1eq123d.3	. . 3 |- ( ph -> C = D )
8		f1eq3 6098	. . 3 |- ( C = D -> ( G : B -1-1-> C <-> G : B -1-1-> D ) )
9	7, 8	syl 17	. 2 |- ( ph -> ( G : B -1-1-> C <-> G : B -1-1-> D ) )
10	3, 6, 9	3bitrd 294	1 |- ( ph -> ( F : A -1-1-> C <-> G : B -1-1-> D ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   -1-1->wf1 5885

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-f 5892  df-f1 5893

This theorem is referenced by:  f10d  6170  fthf1  16577  cofth  16595  istrkgld  25358  istrkg2ld  25359  isushgr  25956  isuspgr  26047  isusgr  26048  isuspgrop  26056  isusgrop  26057  ausgrusgrb  26060  ausgrusgri  26063  usgrstrrepe  26127  uspgr1e  26136  usgrexmpl  26155  usgrres1  26207  usgrexi  26337  uspgr2wlkeq  26542  usgr2trlncl  26656  aciunf1  29463