Theorem foeq123d 6847

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⟶onto𝐶 ↔ 𝐺:𝐵⟶onto𝐷))

Proof of Theorem foeq123d

Step	Hyp	Ref	Expression
1		f1eq123d.1	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
2		foeq1 6822	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶onto𝐶 ↔ 𝐺:𝐴⟶onto𝐶))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → (𝐹:𝐴⟶onto𝐶 ↔ 𝐺:𝐴⟶onto𝐶))
4		f1eq123d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
5		foeq2 6823	. . 3 ⊢ (𝐴 = 𝐵 → (𝐺:𝐴⟶onto𝐶 ↔ 𝐺:𝐵⟶onto𝐶))
6	4, 5	syl 18	. 2 ⊢ (𝜑 → (𝐺:𝐴⟶onto𝐶 ↔ 𝐺:𝐵⟶onto𝐶))
7		f1eq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
8		foeq3 6824	. . 3 ⊢ (𝐶 = 𝐷 → (𝐺:𝐵⟶onto𝐶 ↔ 𝐺:𝐵⟶onto𝐷))
9	7, 8	syl 18	. 2 ⊢ (𝜑 → (𝐺:𝐵⟶onto𝐶 ↔ 𝐺:𝐵⟶onto𝐷))
10	3, 6, 9	3bitrd 308	1 ⊢ (𝜑 → (𝐹:𝐴⟶onto𝐶 ↔ 𝐺:𝐵⟶onto𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ⟶ontowfo 6564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-fun 6568  df-fn 6569  df-fo 6572

This theorem is used by:  fullfo  18139  cofull  18161  resgrpplusfrn  19211  efabl  26935  iseupth  30859  funfocofob  48421  fundcmpsurinjimaid  48766  fundcmpsurinjALT  48767  cofidf2  50567