Theorem f1oeq123d 5357

Description: Equality deduction for one-to-one 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
f1oeq123d	|- ( ph -> ( F : A -1-1-onto-> C <-> G : B -1-1-onto-> D ) )

Proof of Theorem f1oeq123d

Step	Hyp	Ref	Expression
1		f1eq123d.1	. . 3 |- ( ph -> F = G )
2		f1oeq1 5351	. . 3 |- ( F = G -> ( F : A -1-1-onto-> C <-> G : A -1-1-onto-> C ) )
3	1, 2	syl 14	. 2 |- ( ph -> ( F : A -1-1-onto-> C <-> G : A -1-1-onto-> C ) )
4		f1eq123d.2	. . 3 |- ( ph -> A = B )
5		f1oeq2 5352	. . 3 |- ( A = B -> ( G : A -1-1-onto-> C <-> G : B -1-1-onto-> C ) )
6	4, 5	syl 14	. 2 |- ( ph -> ( G : A -1-1-onto-> C <-> G : B -1-1-onto-> C ) )
7		f1eq123d.3	. . 3 |- ( ph -> C = D )
8		f1oeq3 5353	. . 3 |- ( C = D -> ( G : B -1-1-onto-> C <-> G : B -1-1-onto-> D ) )
9	7, 8	syl 14	. 2 |- ( ph -> ( G : B -1-1-onto-> C <-> G : B -1-1-onto-> D ) )
10	3, 6, 9	3bitrd 213	1 |- ( ph -> ( F : A -1-1-onto-> C <-> G : B -1-1-onto-> D ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   -1-1-onto->wf1o 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125

This theorem is referenced by:  f1oprg  5404  ennnfonelemhf1o  11915