Theorem f1oeq123d 5608

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 5602	. . 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 5603	. . 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 5604	. . 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 1398   -1-1-onto->wf1o 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359

This theorem is referenced by:  f1oprg  5660  fzf1o  12061  ennnfonelemhf1o  13164  rhmf1o  14313  ushgredgedg  16221  ushgredgedgloop  16223  trlreslem  16384  gfsumval  16862  gsumgfsumlem  16865  gsumgfsum  16866  gfsumsn  16867