Theorem foeq123d 5122

Description: Equality deduction for 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
foeq123d	|- ( ph -> ( F : A -onto-> C <-> G : B -onto-> D ) )

Proof of Theorem foeq123d

Step	Hyp	Ref	Expression
1		f1eq123d.1	. . 3 |- ( ph -> F = G )
2		foeq1 5102	. . 3 |- ( F = G -> ( F : A -onto-> C <-> G : A -onto-> C ) )
3	1, 2	syl 14	. 2 |- ( ph -> ( F : A -onto-> C <-> G : A -onto-> C ) )
4		f1eq123d.2	. . 3 |- ( ph -> A = B )
5		foeq2 5103	. . 3 |- ( A = B -> ( G : A -onto-> C <-> G : B -onto-> C ) )
6	4, 5	syl 14	. 2 |- ( ph -> ( G : A -onto-> C <-> G : B -onto-> C ) )
7		f1eq123d.3	. . 3 |- ( ph -> C = D )
8		foeq3 5104	. . 3 |- ( C = D -> ( G : B -onto-> C <-> G : B -onto-> D ) )
9	7, 8	syl 14	. 2 |- ( ph -> ( G : B -onto-> C <-> G : B -onto-> D ) )
10	3, 6, 9	3bitrd 203	1 |- ( ph -> ( F : A -onto-> C <-> G : B -onto-> D ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   -onto->wfo 4900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-fo 4908

This theorem is referenced by: (None)