Theorem f1oeq123d 6133

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 6127	. . 3 |- ( F = G -> ( F : A -1-1-onto-> C <-> G : A -1-1-onto-> C ) )
3	1, 2	syl 17	. 2 |- ( ph -> ( F : A -1-1-onto-> C <-> G : A -1-1-onto-> C ) )
4		f1eq123d.2	. . 3 |- ( ph -> A = B )
5		f1oeq2 6128	. . 3 |- ( A = B -> ( G : A -1-1-onto-> C <-> G : B -1-1-onto-> C ) )
6	4, 5	syl 17	. 2 |- ( ph -> ( G : A -1-1-onto-> C <-> G : B -1-1-onto-> C ) )
7		f1eq123d.3	. . 3 |- ( ph -> C = D )
8		f1oeq3 6129	. . 3 |- ( C = D -> ( G : B -1-1-onto-> C <-> G : B -1-1-onto-> D ) )
9	7, 8	syl 17	. 2 |- ( ph -> ( G : B -1-1-onto-> C <-> G : B -1-1-onto-> D ) )
10	3, 6, 9	3bitrd 294	1 |- ( ph -> ( F : A -1-1-onto-> C <-> G : B -1-1-onto-> D ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  f1oprswap  6180  f1oprg  6181  cnfcom  8597  ackbij2lem2  9062  s2f1o  13661  s4f1o  13663  idffth  16593  ressffth  16598  symg1bas  17816  symg2bas  17818  symgfixels  17854  symgfixelsi  17855  rhmf1o  18732  mat1f1o  20284  isismt  25429  ushgredgedg  26121  ushgredgedgloop  26123  trlreslem  26596  wwlksnextbij  26797  eupth0  27074  eupthp1  27076  foresf1o  29343  f1ocnt  29559  indf1ofs  30088  eulerpartgbij  30434  eulerpartlemn  30443  reprpmtf1o  30704  poimirlem16  33425  poimirlem17  33426  poimirlem19  33428  poimirlem20  33429  poimirlem28  33437  wessf1ornlem  39371  disjf1o  39378  ssnnf1octb  39382  sge0fodjrnlem  40633  rnghmf1o  41903