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Theorem f1oabexg 7963
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:𝐴1-1-onto𝐵𝜑)}
Assertion
Ref Expression
f1oabexg ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hints:   𝜑(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝐹(𝑓)

Proof of Theorem f1oabexg
StepHypRef Expression
1 elex 3499 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3499 . 2 (𝐵𝐷𝐵 ∈ V)
3 f1oabexg.1 . . 3 𝐹 = {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)}
4 f1of 6849 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
54ad2antrl 728 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝑓:𝐴1-1-onto𝐵𝜑)) → 𝑓:𝐴𝐵)
6 simpl 482 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
7 simpr 484 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
85, 6, 7fabexd 7958 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ∈ V)
93, 8eqeltrid 2843 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
101, 2, 9syl2an 596 1 ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478  wf 6559  1-1-ontowf1o 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-f1o 6570
This theorem is referenced by: (None)
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