Theorem f1oeq123d 5427

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 5421	. . 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 5422	. . 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 5423	. . 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 1343   -1-1-onto->wf1o 5187

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195

This theorem is referenced by:  f1oprg  5476  ennnfonelemhf1o  12346