Theorem f1oeq123d 6814

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 6808	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐴–1-1-onto→𝐶))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐴–1-1-onto→𝐶))
4		f1eq123d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
5		f1oeq2 6809	. . 3 ⊢ (𝐴 = 𝐵 → (𝐺:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐶))
6	4, 5	syl 18	. 2 ⊢ (𝜑 → (𝐺:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐶))
7		f1eq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
8		f1oeq3 6810	. . 3 ⊢ (𝐶 = 𝐷 → (𝐺:𝐵–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷))
9	7, 8	syl 18	. 2 ⊢ (𝜑 → (𝐺:𝐵–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷))
10	3, 6, 9	3bitrd 308	1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570  –1-1-onto→wf1o 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543

This theorem is referenced by:  f1oprswap  6866  f1oprg  6867  f1ossf1o  7124  cnfcom  9665  ackbij2lem2  10218  idffth  17987  ressffth  17992  symgval  19436  symg1bas  19456  symg2bas  19458  symgfixels  19499  symgfixelsi  19500  rnghmf1o  20530  rhmf1o  20575  mat1f1o  22635  ushgredgedg  29579  ushgredgedgloop  29581  trlreslem  30047  wlknwwlksnbij  30237  wwlksnextbij  30251  clwlknf1oclwwlkn  30435  eupth0  30565  eupthp1  30567  foresf1o  32850  f1ocnt  33145  indf1ofs  33186  gsumwrd2dccat  33398  symgcom  33403  cycpmcl  33436  cycpmconjslem2  33475  nsgqusf1o  33725  1arithidomlem2  33826  1arithidom  33827  dimkerim  34017  eulerpartgbij  34762  eulerpartlemn  34771  reprpmtf1o  35013  poimirlem16  38287  poimirlem17  38288  poimirlem19  38290  poimirlem20  38291  poimirlem28  38299  wessf1ornlem  45903  disjf1o  45909  ssnnf1octb  45912  sge0fodjrnlem  47130  f1oresf1orab  48026  isgrim  48647  isubgrgrim  48694  isgrlim  48747  uspgrlim  48757  grlimedgclnbgr  48760  grlimgrtri  48768  grilcbri2  48776  gpg5grlim  48858  swapf1f1o  50053  swapf2f1o  50054  swapf2f1oa  50055  swapf2f1oaALT  50056  fucoppc  50188