Theorem f1eq123d 5513

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 5475	. . 3 |- ( F = G -> ( F : A -1-1-> C <-> G : A -1-1-> C ) )
3	1, 2	syl 14	. 2 |- ( ph -> ( F : A -1-1-> C <-> G : A -1-1-> C ) )
4		f1eq123d.2	. . 3 |- ( ph -> A = B )
5		f1eq2 5476	. . 3 |- ( A = B -> ( G : A -1-1-> C <-> G : B -1-1-> C ) )
6	4, 5	syl 14	. 2 |- ( ph -> ( G : A -1-1-> C <-> G : B -1-1-> C ) )
7		f1eq123d.3	. . 3 |- ( ph -> C = D )
8		f1eq3 5477	. . 3 |- ( C = D -> ( G : B -1-1-> C <-> G : B -1-1-> D ) )
9	7, 8	syl 14	. 2 |- ( ph -> ( G : B -1-1-> C <-> G : B -1-1-> D ) )
10	3, 6, 9	3bitrd 214	1 |- ( ph -> ( F : A -1-1-> C <-> G : B -1-1-> D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1372   -1-1->wf1 5267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275

This theorem is referenced by: (None)