Theorem f1oabexg 7964

Description: The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.) (Proof shortened by AV, 9-Jun-2025.)

Hypothesis

Ref	Expression
f1oabexg.1	⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)}

Assertion

Ref	Expression
f1oabexg	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Allowed substitution hints:   𝜑(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝐹(𝑓)

Proof of Theorem f1oabexg

Step	Hyp	Ref	Expression
1		elex 3501	. 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		elex 3501	. 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
3		f1oabexg.1	. . 3 ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)}
4		f1of 6848	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
5	4	ad2antrl 728	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)) → 𝑓:𝐴⟶𝐵)
6		simpl 482	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
7		simpr 484	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
8	5, 6, 7	fabexd 7959	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ∈ V)
9	3, 8	eqeltrid 2845	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
10	1, 2, 9	syl2an 596	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  Vcvv 3480  ⟶wf 6557  –1-1-onto→wf1o 6560

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-f1o 6568

This theorem is referenced by: (None)