Theorem f1oabexg 7872

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 3457	. 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		elex 3457	. 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
3		f1oabexg.1	. . 3 ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)}
4		f1of 6763	. . . . 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 7867	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ∈ V)
9	3, 8	eqeltrid 2835	. 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 1541   ∈ wcel 2111  {cab 2709  Vcvv 3436  ⟶wf 6477  –1-1-onto→wf1o 6480

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-f1o 6488

This theorem is referenced by: (None)