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Theorem fsetprcnexALT 47074
Description: First version of proof for fsetprcnex 8902, which was much more complicated. (Contributed by AV, 14-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fsetprcnexALT (((𝐴𝑉𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓𝑓:𝐴𝐵} ∉ V)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem fsetprcnexALT
Dummy variables 𝑎 𝑏 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abanssl 4311 . 2 {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ⊆ {𝑓𝑓:𝐴𝐵}
2 n0 4353 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
3 vex 3484 . . . . . . . . . . . 12 𝑦 ∈ V
43a1i 11 . . . . . . . . . . 11 ((𝑦𝐴𝐴𝑉) → 𝑦 ∈ V)
5 fsetsnprcnex 47067 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ 𝐵 ∉ V) → {𝑓𝑓:{𝑦}⟶𝐵} ∉ V)
64, 5sylan 580 . . . . . . . . . 10 (((𝑦𝐴𝐴𝑉) ∧ 𝐵 ∉ V) → {𝑓𝑓:{𝑦}⟶𝐵} ∉ V)
7 df-nel 3047 . . . . . . . . . 10 ({𝑓𝑓:{𝑦}⟶𝐵} ∉ V ↔ ¬ {𝑓𝑓:{𝑦}⟶𝐵} ∈ V)
86, 7sylib 218 . . . . . . . . 9 (((𝑦𝐴𝐴𝑉) ∧ 𝐵 ∉ V) → ¬ {𝑓𝑓:{𝑦}⟶𝐵} ∈ V)
9 eqid 2737 . . . . . . . . . . . . 13 {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} = {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)}
10 eqid 2737 . . . . . . . . . . . . 13 {𝑓𝑓:{𝑦}⟶𝐵} = {𝑓𝑓:{𝑦}⟶𝐵}
11 eqid 2737 . . . . . . . . . . . . 13 (𝑔 ∈ {𝑓𝑓:{𝑦}⟶𝐵} ↦ (𝑎𝐴 ↦ (𝑔𝑦))) = (𝑔 ∈ {𝑓𝑓:{𝑦}⟶𝐵} ↦ (𝑎𝐴 ↦ (𝑔𝑦)))
129, 10, 11cfsetsnfsetf1o 47073 . . . . . . . . . . . 12 ((𝐴𝑉𝑦𝐴) → (𝑔 ∈ {𝑓𝑓:{𝑦}⟶𝐵} ↦ (𝑎𝐴 ↦ (𝑔𝑦))):{𝑓𝑓:{𝑦}⟶𝐵}–1-1-onto→{𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)})
1312ancoms 458 . . . . . . . . . . 11 ((𝑦𝐴𝐴𝑉) → (𝑔 ∈ {𝑓𝑓:{𝑦}⟶𝐵} ↦ (𝑎𝐴 ↦ (𝑔𝑦))):{𝑓𝑓:{𝑦}⟶𝐵}–1-1-onto→{𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)})
1413adantr 480 . . . . . . . . . 10 (((𝑦𝐴𝐴𝑉) ∧ 𝐵 ∉ V) → (𝑔 ∈ {𝑓𝑓:{𝑦}⟶𝐵} ↦ (𝑎𝐴 ↦ (𝑔𝑦))):{𝑓𝑓:{𝑦}⟶𝐵}–1-1-onto→{𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)})
15 f1ovv 7982 . . . . . . . . . . 11 ((𝑔 ∈ {𝑓𝑓:{𝑦}⟶𝐵} ↦ (𝑎𝐴 ↦ (𝑔𝑦))):{𝑓𝑓:{𝑦}⟶𝐵}–1-1-onto→{𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} → ({𝑓𝑓:{𝑦}⟶𝐵} ∈ V ↔ {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∈ V))
1615bicomd 223 . . . . . . . . . 10 ((𝑔 ∈ {𝑓𝑓:{𝑦}⟶𝐵} ↦ (𝑎𝐴 ↦ (𝑔𝑦))):{𝑓𝑓:{𝑦}⟶𝐵}–1-1-onto→{𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} → ({𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∈ V ↔ {𝑓𝑓:{𝑦}⟶𝐵} ∈ V))
1714, 16syl 17 . . . . . . . . 9 (((𝑦𝐴𝐴𝑉) ∧ 𝐵 ∉ V) → ({𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∈ V ↔ {𝑓𝑓:{𝑦}⟶𝐵} ∈ V))
188, 17mtbird 325 . . . . . . . 8 (((𝑦𝐴𝐴𝑉) ∧ 𝐵 ∉ V) → ¬ {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∈ V)
1918exp31 419 . . . . . . 7 (𝑦𝐴 → (𝐴𝑉 → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∈ V)))
2019exlimiv 1930 . . . . . 6 (∃𝑦 𝑦𝐴 → (𝐴𝑉 → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∈ V)))
212, 20sylbi 217 . . . . 5 (𝐴 ≠ ∅ → (𝐴𝑉 → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∈ V)))
2221impcom 407 . . . 4 ((𝐴𝑉𝐴 ≠ ∅) → (𝐵 ∉ V → ¬ {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∈ V))
2322imp 406 . . 3 (((𝐴𝑉𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → ¬ {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∈ V)
24 df-nel 3047 . . 3 ({𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∉ V ↔ ¬ {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∈ V)
2523, 24sylibr 234 . 2 (((𝐴𝑉𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∉ V)
26 prcssprc 5327 . 2 (({𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ⊆ {𝑓𝑓:𝐴𝐵} ∧ {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)} ∉ V) → {𝑓𝑓:𝐴𝐵} ∉ V)
271, 25, 26sylancr 587 1 (((𝐴𝑉𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓𝑓:𝐴𝐵} ∉ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wnel 3046  wral 3061  wrex 3070  Vcvv 3480  wss 3951  c0 4333  {csn 4626  cmpt 5225  wf 6557  1-1-ontowf1o 6560  cfv 6561
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by: (None)
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