Theorem f1oeq123d 6386

Description: Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
f1eq123d.1	⊢ (𝜑 → 𝐹 = 𝐺)
f1eq123d.2	⊢ (𝜑 → 𝐴 = 𝐵)
f1eq123d.3	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
f1oeq123d	⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷))

Proof of Theorem f1oeq123d

Step	Hyp	Ref	Expression
1		f1eq123d.1	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
2		f1oeq1 6380	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐴–1-1-onto→𝐶))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐴–1-1-onto→𝐶))
4		f1eq123d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
5		f1oeq2 6381	. . 3 ⊢ (𝐴 = 𝐵 → (𝐺:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐶))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (𝐺:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐶))
7		f1eq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
8		f1oeq3 6382	. . 3 ⊢ (𝐶 = 𝐷 → (𝐺:𝐵–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷))
9	7, 8	syl 17	. 2 ⊢ (𝜑 → (𝐺:𝐵–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷))
10	3, 6, 9	3bitrd 297	1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1601  –1-1-onto→wf1o 6134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142

This theorem is referenced by:  f1oprswap  6434  f1oprg  6435  f1ossf1o  6660  cnfcom  8894  ackbij2lem2  9397  s2f1o  14067  s4f1o  14069  idffth  16978  ressffth  16983  symg1bas  18199  symg2bas  18201  symgfixels  18237  symgfixelsi  18238  rhmf1o  19121  mat1f1o  20689  isismt  25885  ushgredgedg  26575  ushgredgedgloop  26577  ushgredgedgloopOLD  26578  trlreslem  27050  trlreslemOLD  27052  wlknwwlksnbij  27242  wwlksnextbij  27271  wwlksnextbijOLD  27272  clwlknf1oclwwlkn  27483  clwlknf1oclwwlknOLD  27485  eupth0  27617  eupthp1  27620  foresf1o  29905  f1ocnt  30123  dimkerim  30441  indf1ofs  30686  eulerpartgbij  31032  eulerpartlemn  31041  reprpmtf1o  31306  poimirlem16  34051  poimirlem17  34052  poimirlem19  34054  poimirlem20  34055  poimirlem28  34063  wessf1ornlem  40294  disjf1o  40301  ssnnf1octb  40305  sge0fodjrnlem  41557  f1oresf1orab  42330  isomgr  42736  rnghmf1o  42918