Theorem foeq123d 6775

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 6750	. . 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 6751	. . 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 6752	. . 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 1542  –onto→wfo 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-fun 6502  df-fn 6503  df-fo 6506

This theorem is referenced by:  fullfo  17850  cofull  17872  resgrpplusfrn  18892  efabl  26527  iseupth  30288  funfocofob  47438  fundcmpsurinjimaid  47771  fundcmpsurinjALT  47772  cofidf2  49479