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Theorem fvmptopab1 6594
 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.)
Hypotheses
Ref Expression
fvmptopab1.2 ((𝜑𝑧 = 𝑍) → (𝜒𝜓))
fvmptopab1.3 (𝜑𝑍𝑉)
fvmptopab1.4 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
fvmptopab1.1 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)})
Assertion
Ref Expression
fvmptopab1 (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)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem fvmptopab1
StepHypRef Expression
1 fvmptopab1.1 . . 3 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)})
21a1i 11 . 2 (𝜑𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)}))
3 fveq2 6103 . . . . . 6 (𝑧 = 𝑍 → (𝐹𝑧) = (𝐹𝑍))
43breqd 4594 . . . . 5 (𝑧 = 𝑍 → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
54adantl 481 . . . 4 ((𝜑𝑧 = 𝑍) → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
6 fvmptopab1.2 . . . 4 ((𝜑𝑧 = 𝑍) → (𝜒𝜓))
75, 6anbi12d 743 . . 3 ((𝜑𝑧 = 𝑍) → ((𝑥(𝐹𝑧)𝑦𝜒) ↔ (𝑥(𝐹𝑍)𝑦𝜓)))
87opabbidv 4648 . 2 ((𝜑𝑧 = 𝑍) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
9 fvmptopab1.3 . . 3 (𝜑𝑍𝑉)
109elexd 3187 . 2 (𝜑𝑍 ∈ V)
11 id 22 . . . 4 (𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
1211gen2 1714 . . 3 𝑥𝑦(𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
13 fvmptopab1.4 . . 3 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
14 opabbrex 6593 . . 3 ((∀𝑥𝑦(𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V)
1512, 13, 14sylancr 694 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V)
162, 8, 10, 15fvmptd 6197 1 (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  fvmptopab2  6595  trlsfval  40903  spthsfval  40928  clwlkS  40978  crctS  40994  cyclS  40995  eupths  41367
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