Theorem f1oeq123d 6816

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 6810	. . 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 6811	. . 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 6812	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  –1-1-onto→wf1o 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544

This theorem is used by:  f1oprswap  6868  f1oprg  6869  f1ossf1o  7127  cnfcom  9694  ackbij2lem2  10310  idffth  18103  ressffth  18108  symgval  19578  symg1bas  19598  symg2bas  19600  symgfixels  19641  symgfixelsi  19642  rnghmf1o  20675  rhmf1o  20720  mat1f1o  22786  ushgredgedg  29803  ushgredgedgloop  29805  trlreslem  30275  wlknwwlksnbij  30470  wwlksnextbij  30484  clwlknf1oclwwlkn  30668  eupth0  30808  eupthp1  30810  foresf1o  33093  f1ocnt  33385  indf1ofs  33426  gsumwrd2dccat  33632  symgcom  33637  cycpmcl  33670  cycpmconjslem2  33709  nsgqusf1o  33960  1arithidomlem2  34061  1arithidom  34062  dimkerim  34252  eulerpartgbij  34997  eulerpartlemn  35006  reprpmtf1o  35248  poimirlem16  38534  poimirlem17  38535  poimirlem19  38537  poimirlem20  38538  poimirlem28  38546  wessf1ornlem  46169  disjf1o  46175  ssnnf1octb  46178  sge0fodjrnlem  47395  f1oresf1orab  48328  isgrim  48949  isubgrgrim  48996  isgrlim  49049  uspgrlim  49059  grlimedgclnbgr  49062  grlimgrtri  49070  grilcbri2  49078  gpg5grlim  49160  swapf1f1o  50352  swapf2f1o  50353  swapf2f1oa  50354  swapf2f1oaALT  50355  fucoppc  50487