MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1oabexgOLD Structured version   Visualization version   GIF version

Theorem f1oabexgOLD 7965
Description: Obsolete version of f1oabexg 7964 as of 9-Jun-2025. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
f1oabexg.1 𝐹 = {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)}
Assertion
Ref Expression
f1oabexgOLD ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hints:   𝜑(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝐹(𝑓)

Proof of Theorem f1oabexgOLD
StepHypRef Expression
1 f1oabexg.1 . 2 𝐹 = {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)}
2 f1of 6848 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
32anim1i 615 . . . 4 ((𝑓:𝐴1-1-onto𝐵𝜑) → (𝑓:𝐴𝐵𝜑))
43ss2abi 4067 . . 3 {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)}
5 eqid 2737 . . . 4 {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} = {𝑓 ∣ (𝑓:𝐴𝐵𝜑)}
65fabexg 7960 . . 3 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∈ V)
7 ssexg 5323 . . 3 (({𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ⊆ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∧ {𝑓 ∣ (𝑓:𝐴𝐵𝜑)} ∈ V) → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ∈ V)
84, 6, 7sylancr 587 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐵𝜑)} ∈ V)
91, 8eqeltrid 2845 1 ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  wss 3951  wf 6557  1-1-ontowf1o 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-f1o 6568
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator