Theorem f1oeq123d 6827

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 6821	. . 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 6822	. . 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 6823	. . 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 304	1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541  –1-1-onto→wf1o 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550

This theorem is referenced by:  f1oprswap  6877  f1oprg  6878  f1ossf1o  7128  cnfcom  9697  ackbij2lem2  10237  idffth  17886  ressffth  17891  symgval  19238  symg1bas  19260  symg2bas  19262  symgfixels  19304  symgfixelsi  19305  rhmf1o  20273  mat1f1o  21987  ushgredgedg  28524  ushgredgedgloop  28526  trlreslem  28994  wlknwwlksnbij  29180  wwlksnextbij  29194  clwlknf1oclwwlkn  29375  eupth0  29505  eupthp1  29507  foresf1o  31780  f1ocnt  32051  symgcom  32285  cycpmcl  32316  cycpmconjslem2  32355  nsgqusf1o  32572  dimkerim  32771  indf1ofs  33093  eulerpartgbij  33440  eulerpartlemn  33449  reprpmtf1o  33707  poimirlem16  36590  poimirlem17  36591  poimirlem19  36593  poimirlem20  36594  poimirlem28  36602  metakunt17  41087  wessf1ornlem  43963  disjf1o  43969  ssnnf1octb  43972  sge0fodjrnlem  45211  f1oresf1orab  46076  isomgr  46570  rnghmf1o  46780