Theorem f1oeq123d 6795

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 6789	. . 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 6790	. . 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 6791	. . 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 307	1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559  –1-1-onto→wf1o 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523

This theorem is referenced by:  f1oprswap  6847  f1oprg  6848  f1ossf1o  7105  cnfcom  9649  ackbij2lem2  10189  idffth  17959  ressffth  17964  symgval  19402  symg1bas  19422  symg2bas  19424  symgfixels  19465  symgfixelsi  19466  rnghmf1o  20488  rhmf1o  20527  mat1f1o  22526  ushgredgedg  29387  ushgredgedgloop  29389  trlreslem  29855  wlknwwlksnbij  30045  wwlksnextbij  30059  clwlknf1oclwwlkn  30243  eupth0  30373  eupthp1  30375  foresf1o  32663  f1ocnt  32963  indf1ofs  33005  gsumwrd2dccat  33219  symgcom  33224  cycpmcl  33257  cycpmconjslem2  33296  nsgqusf1o  33563  1arithidomlem2  33693  1arithidom  33694  dimkerim  33885  eulerpartgbij  34630  eulerpartlemn  34639  reprpmtf1o  34881  poimirlem16  38096  poimirlem17  38097  poimirlem19  38099  poimirlem20  38100  poimirlem28  38108  wessf1ornlem  45724  disjf1o  45730  ssnnf1octb  45733  sge0fodjrnlem  46951  f1oresf1orab  47844  isgrim  48465  isubgrgrim  48512  isgrlim  48565  uspgrlim  48575  grlimedgclnbgr  48578  grlimgrtri  48586  grilcbri2  48594  gpg5grlim  48676  swapf1f1o  49857  swapf2f1o  49858  swapf2f1oa  49859  swapf2f1oaALT  49860  fucoppc  49992