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Theorem fvmptopab 6931
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 6411 . . . . . . . 8 (𝑧 = 𝑍 → (𝐹𝑧) = (𝐹𝑍))
43breqd 4854 . . . . . . 7 (𝑧 = 𝑍 → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
54adantl 474 . . . . . 6 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
6 fvmptopab.1 . . . . . . 7 ((𝜑𝑧 = 𝑍) → (𝜒𝜓))
76adantll 706 . . . . . 6 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝜒𝜓))
85, 7anbi12d 625 . . . . 5 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → ((𝑥(𝐹𝑧)𝑦𝜒) ↔ (𝑥(𝐹𝑍)𝑦𝜓)))
98opabbidv 4909 . . . 4 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
10 simpl 475 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
11 id 22 . . . . . 6 (𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
1211gen2 1892 . . . . 5 𝑥𝑦(𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦)
13 fvmptopab.2 . . . . . 6 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
1413adantl 474 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V)
15 opabbrex 6929 . . . . 5 ((∀𝑥𝑦(𝑥(𝐹𝑍)𝑦𝑥(𝐹𝑍)𝑦) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹𝑍)𝑦} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V)
1612, 14, 15sylancr 582 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V)
172, 9, 10, 16fvmptd 6513 . . 3 ((𝑍 ∈ V ∧ 𝜑) → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
1817ex 402 . 2 (𝑍 ∈ V → (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)}))
19 fvprc 6404 . . . 4 𝑍 ∈ V → (𝑀𝑍) = ∅)
20 br0 4892 . . . . . . . 8 ¬ 𝑥𝑦
21 fvprc 6404 . . . . . . . . 9 𝑍 ∈ V → (𝐹𝑍) = ∅)
2221breqd 4854 . . . . . . . 8 𝑍 ∈ V → (𝑥(𝐹𝑍)𝑦𝑥𝑦))
2320, 22mtbiri 319 . . . . . . 7 𝑍 ∈ V → ¬ 𝑥(𝐹𝑍)𝑦)
2423intnanrd 484 . . . . . 6 𝑍 ∈ V → ¬ (𝑥(𝐹𝑍)𝑦𝜓))
2524alrimivv 2024 . . . . 5 𝑍 ∈ V → ∀𝑥𝑦 ¬ (𝑥(𝐹𝑍)𝑦𝜓))
26 opab0 5203 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥(𝐹𝑍)𝑦𝜓))
2725, 26sylibr 226 . . . 4 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅)
2819, 27eqtr4d 2836 . . 3 𝑍 ∈ V → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
2928a1d 25 . 2 𝑍 ∈ V → (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)}))
3018, 29pm2.61i 177 1 (𝜑 → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wal 1651   = wceq 1653  wcel 2157  Vcvv 3385  c0 4115   class class class wbr 4843  {copab 4905  cmpt 4922  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109
This theorem is referenced by:  trlsfval  26948  pthsfval  26975  spthsfval  26976  clwlks  27026  crcts  27042  cycls  27043  eupths  27544
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