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Theorem f1oabexg 5468
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 5456 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
32anim1i 340 . . . 4 ((𝑓:𝐴1-1-onto𝐵𝜑) → (𝑓:𝐴𝐵𝜑))
43ss2abi 3227 . . 3 {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)}
5 eqid 2177 . . . 4 {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} = {𝑓 ∣ (𝑓:𝐴𝐵𝜑)}
65fabexg 5398 . . 3 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∈ V)
7 ssexg 4139 . . 3 (({𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∧ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∈ V) → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ∈ V)
84, 6, 7sylancr 414 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ∈ V)
91, 8eqeltrid 2264 1 ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  {cab 2163  Vcvv 2737  wss 3129  wf 5207  1-1-ontowf1o 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-dm 4632  df-rn 4633  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-f1o 5218
This theorem is referenced by: (None)
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