Theorem f1oeq123d 5467

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 5461	. . 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 5462	. . 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 5463	. . 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 214	1 |- ( ph -> ( F : A -1-1-onto-> C <-> G : B -1-1-onto-> D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1363   -1-1-onto->wf1o 5227

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235

This theorem is referenced by:  f1oprg  5517  ennnfonelemhf1o  12428