MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1oabexg Structured version   Visualization version   GIF version

Theorem f1oabexg 7271
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 6278 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
32anim1i 594 . . . 4 ((𝑓:𝐴1-1-onto𝐵𝜑) → (𝑓:𝐴𝐵𝜑))
43ss2abi 3821 . . 3 {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)}
5 eqid 2770 . . . 4 {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} = {𝑓 ∣ (𝑓:𝐴𝐵𝜑)}
65fabexg 7268 . . 3 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∈ V)
7 ssexg 4935 . . 3 (({𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∧ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∈ V) → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ∈ V)
84, 6, 7sylancr 567 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ∈ V)
91, 8syl5eqel 2853 1 ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  {cab 2756  Vcvv 3349  wss 3721  wf 6027  1-1-ontowf1o 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-f1o 6038
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator