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Theorem fvmptopabOLD 7488
Description: Obsolete version of fvmptopab 7487 as of 13-Dec-2024. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
fvmptopabOLD.1 ((𝜑𝑧 = 𝑍) → (𝜒𝜓))
fvmptopabOLD.2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
fvmptopabOLD.3 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)})
Assertion
Ref Expression
fvmptopabOLD (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
Distinct variable groups:   𝑧,𝐹   𝑥,𝑍,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝜓,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦)   𝑀(𝑥,𝑦,𝑧)

Proof of Theorem fvmptopabOLD
StepHypRef Expression
1 fvmptopabOLD.3 . . . 4 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)})
2 fveq2 6907 . . . . . . . 8 (𝑧 = 𝑍 → (𝐹𝑧) = (𝐹𝑍))
32breqd 5159 . . . . . . 7 (𝑧 = 𝑍 → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
43adantl 481 . . . . . 6 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
5 fvmptopabOLD.1 . . . . . . 7 ((𝜑𝑧 = 𝑍) → (𝜒𝜓))
65adantll 714 . . . . . 6 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝜒𝜓))
74, 6anbi12d 632 . . . . 5 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → ((𝑥(𝐹𝑧)𝑦𝜒) ↔ (𝑥(𝐹𝑍)𝑦𝜓)))
87opabbidv 5214 . . . 4 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
9 simpl 482 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
10 id 22 . . . . . 6 (𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
1110gen2 1793 . . . . 5 𝑥𝑦(𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
12 fvmptopabOLD.2 . . . . . 6 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
1312adantl 481 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
14 opabbrex 7484 . . . . 5 ((∀𝑥𝑦(𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V)
1511, 13, 14sylancr 587 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V)
161, 8, 9, 15fvmptd2 7024 . . 3 ((𝑍 ∈ V ∧ 𝜑) → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
1716ex 412 . 2 (𝑍 ∈ V → (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)}))
18 fvprc 6899 . . . 4 𝑍 ∈ V → (𝑀𝑍) = ∅)
19 br0 5197 . . . . . . . 8 ¬ 𝑥𝑦
20 fvprc 6899 . . . . . . . . 9 𝑍 ∈ V → (𝐹𝑍) = ∅)
2120breqd 5159 . . . . . . . 8 𝑍 ∈ V → (𝑥(𝐹𝑍)𝑦𝑥𝑦))
2219, 21mtbiri 327 . . . . . . 7 𝑍 ∈ V → ¬ 𝑥(𝐹𝑍)𝑦)
2322intnanrd 489 . . . . . 6 𝑍 ∈ V → ¬ (𝑥(𝐹𝑍)𝑦𝜓))
2423alrimivv 1926 . . . . 5 𝑍 ∈ V → ∀𝑥𝑦 ¬ (𝑥(𝐹𝑍)𝑦𝜓))
25 opab0 5564 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥(𝐹𝑍)𝑦𝜓))
2624, 25sylibr 234 . . . 4 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅)
2718, 26eqtr4d 2778 . . 3 𝑍 ∈ V → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
2827a1d 25 . 2 𝑍 ∈ V → (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)}))
2917, 28pm2.61i 182 1 (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339   class class class wbr 5148  {copab 5210  cmpt 5231  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by: (None)
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