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Theorem fvmptopab 7487
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.) Add disjoint variable condition on 𝐹, 𝑥, 𝑦 to remove a sethood hypothesis. (Revised by SN, 13-Dec-2024.)
Hypotheses
Ref Expression
fvmptopab.1 (𝑧 = 𝑍 → (𝜑𝜓))
fvmptopab.m 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜑)})
Assertion
Ref Expression
fvmptopab (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)}
Distinct variable groups:   𝑥,𝐹,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦)   𝑀(𝑥,𝑦,𝑧)

Proof of Theorem fvmptopab
StepHypRef Expression
1 fveq2 6906 . . . . . 6 (𝑧 = 𝑍 → (𝐹𝑧) = (𝐹𝑍))
21breqd 5154 . . . . 5 (𝑧 = 𝑍 → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
3 fvmptopab.1 . . . . 5 (𝑧 = 𝑍 → (𝜑𝜓))
42, 3anbi12d 632 . . . 4 (𝑧 = 𝑍 → ((𝑥(𝐹𝑧)𝑦𝜑) ↔ (𝑥(𝐹𝑍)𝑦𝜓)))
54opabbidv 5209 . . 3 (𝑧 = 𝑍 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜑)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
6 fvmptopab.m . . 3 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑧)𝑦𝜑)})
7 opabresex2 7485 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V
85, 6, 7fvmpt 7016 . 2 (𝑍 ∈ V → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
9 fvprc 6898 . . 3 𝑍 ∈ V → (𝑀𝑍) = ∅)
10 elopabran 5567 . . . . . 6 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} → 𝑧 ∈ (𝐹𝑍))
1110ssriv 3987 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ⊆ (𝐹𝑍)
12 fvprc 6898 . . . . 5 𝑍 ∈ V → (𝐹𝑍) = ∅)
1311, 12sseqtrid 4026 . . . 4 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ⊆ ∅)
14 ss0 4402 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ⊆ ∅ → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅)
1513, 14syl 17 . . 3 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅)
169, 15eqtr4d 2780 . 2 𝑍 ∈ V → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
178, 16pm2.61i 182 1 (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  c0 4333   class class class wbr 5143  {copab 5205  cmpt 5225  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by:  trlsfval  29713  pthsfval  29739  spthsfval  29740  clwlks  29792  crcts  29808  cycls  29809  eupths  30219
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