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Theorem fvmptopab 6682
 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 . . . . 5 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)})
21a1i 11 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)}))
3 fveq2 6178 . . . . . . . 8 (𝑧 = 𝑍 → (𝐹𝑧) = (𝐹𝑍))
43breqd 4655 . . . . . . 7 (𝑧 = 𝑍 → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
54adantl 482 . . . . . 6 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
6 fvmptopab.1 . . . . . . 7 ((𝜑𝑧 = 𝑍) → (𝜒𝜓))
76adantll 749 . . . . . 6 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝜒𝜓))
85, 7anbi12d 746 . . . . 5 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → ((𝑥(𝐹𝑧)𝑦𝜒) ↔ (𝑥(𝐹𝑍)𝑦𝜓)))
98opabbidv 4707 . . . 4 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
10 simpl 473 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
11 id 22 . . . . . 6 (𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
1211gen2 1721 . . . . 5 𝑥𝑦(𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
13 fvmptopab.2 . . . . . 6 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
1413adantl 482 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
15 opabbrex 6680 . . . . 5 ((∀𝑥𝑦(𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V)
1612, 14, 15sylancr 694 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V)
172, 9, 10, 16fvmptd 6275 . . 3 ((𝑍 ∈ V ∧ 𝜑) → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
1817ex 450 . 2 (𝑍 ∈ V → (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)}))
19 fvprc 6172 . . . 4 𝑍 ∈ V → (𝑀𝑍) = ∅)
20 br0 4692 . . . . . . . 8 ¬ 𝑥𝑦
21 fvprc 6172 . . . . . . . . 9 𝑍 ∈ V → (𝐹𝑍) = ∅)
2221breqd 4655 . . . . . . . 8 𝑍 ∈ V → (𝑥(𝐹𝑍)𝑦𝑥𝑦))
2320, 22mtbiri 317 . . . . . . 7 𝑍 ∈ V → ¬ 𝑥(𝐹𝑍)𝑦)
2423intnanrd 962 . . . . . 6 𝑍 ∈ V → ¬ (𝑥(𝐹𝑍)𝑦𝜓))
2524alrimivv 1854 . . . . 5 𝑍 ∈ V → ∀𝑥𝑦 ¬ (𝑥(𝐹𝑍)𝑦𝜓))
26 opab0 4997 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥(𝐹𝑍)𝑦𝜓))
2725, 26sylibr 224 . . . 4 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅)
2819, 27eqtr4d 2657 . . 3 𝑍 ∈ V → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
2928a1d 25 . 2 𝑍 ∈ V → (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)}))
3018, 29pm2.61i 176 1 (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1479   = wceq 1481   ∈ wcel 1988  Vcvv 3195  ∅c0 3907   class class class wbr 4644  {copab 4703   ↦ cmpt 4720  ‘cfv 5876 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884 This theorem is referenced by:  trlsfval  26573  pthsfval  26598  spthsfval  26599  clwlks  26649  crcts  26664  cycls  26665  eupths  27040
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