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Theorem f1oabexg 7874
Description: The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
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 f1oabexg.1 . 2 𝐹 = {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)}
2 f1of 6785 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
32anim1i 616 . . . 4 ((𝑓:𝐴1-1-onto𝐵𝜑) → (𝑓:𝐴𝐵𝜑))
43ss2abi 4024 . . 3 {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)}
5 eqid 2733 . . . 4 {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} = {𝑓 ∣ (𝑓:𝐴𝐵𝜑)}
65fabexg 7872 . . 3 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∈ V)
7 ssexg 5281 . . 3 (({𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∧ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∈ V) → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ∈ V)
84, 6, 7sylancr 588 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ∈ V)
91, 8eqeltrid 2838 1 ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  Vcvv 3444  wss 3911  wf 6493  1-1-ontowf1o 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-f1o 6504
This theorem is referenced by: (None)
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