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Theorem fvmptopab 7320
Description: The function value of a mapping 𝑀 to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function 𝐹 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.)
Hypotheses
Ref Expression
fvmptopab.1 ((𝜑𝑧 = 𝑍) → (𝜒𝜓))
fvmptopab.2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
fvmptopab.3 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)})
Assertion
Ref Expression
fvmptopab (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 fvmptopab
StepHypRef Expression
1 fvmptopab.3 . . . 4 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)})
2 fveq2 6766 . . . . . . . 8 (𝑧 = 𝑍 → (𝐹𝑧) = (𝐹𝑍))
32breqd 5084 . . . . . . 7 (𝑧 = 𝑍 → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
43adantl 482 . . . . . 6 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
5 fvmptopab.1 . . . . . . 7 ((𝜑𝑧 = 𝑍) → (𝜒𝜓))
65adantll 711 . . . . . 6 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝜒𝜓))
74, 6anbi12d 631 . . . . 5 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → ((𝑥(𝐹𝑧)𝑦𝜒) ↔ (𝑥(𝐹𝑍)𝑦𝜓)))
87opabbidv 5139 . . . 4 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
9 simpl 483 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
10 id 22 . . . . . 6 (𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
1110gen2 1799 . . . . 5 𝑥𝑦(𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
12 fvmptopab.2 . . . . . 6 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
1312adantl 482 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
14 opabbrex 7318 . . . . 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 6875 . . 3 ((𝑍 ∈ V ∧ 𝜑) → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
1716ex 413 . 2 (𝑍 ∈ V → (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)}))
18 fvprc 6758 . . . 4 𝑍 ∈ V → (𝑀𝑍) = ∅)
19 br0 5122 . . . . . . . 8 ¬ 𝑥𝑦
20 fvprc 6758 . . . . . . . . 9 𝑍 ∈ V → (𝐹𝑍) = ∅)
2120breqd 5084 . . . . . . . 8 𝑍 ∈ V → (𝑥(𝐹𝑍)𝑦𝑥𝑦))
2219, 21mtbiri 327 . . . . . . 7 𝑍 ∈ V → ¬ 𝑥(𝐹𝑍)𝑦)
2322intnanrd 490 . . . . . 6 𝑍 ∈ V → ¬ (𝑥(𝐹𝑍)𝑦𝜓))
2423alrimivv 1931 . . . . 5 𝑍 ∈ V → ∀𝑥𝑦 ¬ (𝑥(𝐹𝑍)𝑦𝜓))
25 opab0 5464 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥(𝐹𝑍)𝑦𝜓))
2624, 25sylibr 233 . . . 4 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅)
2718, 26eqtr4d 2781 . . 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 205  wa 396  wal 1537   = wceq 1539  wcel 2106  Vcvv 3429  c0 4256   class class class wbr 5073  {copab 5135  cmpt 5156  cfv 6426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-iota 6384  df-fun 6428  df-fv 6434
This theorem is referenced by:  trlsfval  28072  pthsfval  28097  spthsfval  28098  clwlks  28148  crcts  28164  cycls  28165  eupths  28572
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