Theorem foeq123d 6796

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 6771	. . 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 6772	. . 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 6773	. . 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 305	1 ⊢ (𝜑 → (𝐹:𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  –onto→wfo 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-fun 6516  df-fn 6517  df-fo 6520

This theorem is referenced by:  fullfo  17883  cofull  17905  resgrpplusfrn  18889  efabl  26466  iseupth  30137  funfocofob  47083  fundcmpsurinjimaid  47416  fundcmpsurinjALT  47417  cofidf2  49113