Theorem foeq123d 6693

Description: Equality deduction for 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
foeq123d	⊢ (𝜑 → (𝐹:𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐷))

Proof of Theorem foeq123d

Step	Hyp	Ref	Expression
1		f1eq123d.1	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
2		foeq1 6668	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐶 ↔ 𝐺:𝐴–onto→𝐶))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝐹:𝐴–onto→𝐶 ↔ 𝐺:𝐴–onto→𝐶))
4		f1eq123d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
5		foeq2 6669	. . 3 ⊢ (𝐴 = 𝐵 → (𝐺:𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (𝐺:𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))
7		f1eq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
8		foeq3 6670	. . 3 ⊢ (𝐶 = 𝐷 → (𝐺:𝐵–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐷))
9	7, 8	syl 17	. 2 ⊢ (𝜑 → (𝐺:𝐵–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐷))
10	3, 6, 9	3bitrd 304	1 ⊢ (𝜑 → (𝐹:𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  –onto→wfo 6416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-fun 6420  df-fn 6421  df-fo 6424

This theorem is referenced by:  fullfo  17544  cofull  17566  resgrpplusfrn  18508  efabl  25611  iseupth  28466  funfocofob  44457  fundcmpsurinjimaid  44751  fundcmpsurinjALT  44752