Theorem foeq123d 6611

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 6588	. . 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 6589	. . 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 6590	. . 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 307	1 ⊢ (𝜑 → (𝐹:𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537  –onto→wfo 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-fun 6359  df-fn 6360  df-fo 6363

This theorem is referenced by:  fullfo  17184  cofull  17206  resgrpplusfrn  18119  efabl  25136  iseupth  27982  fundcmpsurinjimaid  43578  fundcmpsurinjALT  43579