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Theorem f1oabexg 7923
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 6826 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
32anim1i 614 . . . 4 ((𝑓:𝐴1-1-onto𝐵𝜑) → (𝑓:𝐴𝐵𝜑))
43ss2abi 4058 . . 3 {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)}
5 eqid 2726 . . . 4 {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} = {𝑓 ∣ (𝑓:𝐴𝐵𝜑)}
65fabexg 7921 . . 3 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∈ V)
7 ssexg 5316 . . 3 (({𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∧ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∈ V) → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ∈ V)
84, 6, 7sylancr 586 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ∈ V)
91, 8eqeltrid 2831 1 ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {cab 2703  Vcvv 3468  wss 3943  wf 6532  1-1-ontowf1o 6535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-f1o 6543
This theorem is referenced by: (None)
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