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Theorem fvmptopab 7474
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 6896 . . . . . 6 (𝑧 = 𝑍 → (𝐹𝑧) = (𝐹𝑍))
21breqd 5160 . . . . 5 (𝑧 = 𝑍 → (𝑥(𝐹𝑧)𝑦𝑥(𝐹𝑍)𝑦))
3 fvmptopab.1 . . . . 5 (𝑧 = 𝑍 → (𝜑𝜓))
42, 3anbi12d 630 . . . 4 (𝑧 = 𝑍 → ((𝑥(𝐹𝑧)𝑦𝜑) ↔ (𝑥(𝐹𝑍)𝑦𝜓)))
54opabbidv 5215 . . 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 7472 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ∈ V
85, 6, 7fvmpt 7004 . 2 (𝑍 ∈ V → (𝑀𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)})
9 fvprc 6888 . . 3 𝑍 ∈ V → (𝑀𝑍) = ∅)
10 elopabran 5564 . . . . . 6 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} → 𝑧 ∈ (𝐹𝑍))
1110ssriv 3980 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ⊆ (𝐹𝑍)
12 fvprc 6888 . . . . 5 𝑍 ∈ V → (𝐹𝑍) = ∅)
1311, 12sseqtrid 4029 . . . 4 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ⊆ ∅)
14 ss0 4400 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} ⊆ ∅ → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅)
1513, 14syl 17 . . 3 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹𝑍)𝑦𝜓)} = ∅)
169, 15eqtr4d 2768 . 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 205  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  wss 3944  c0 4322   class class class wbr 5149  {copab 5211  cmpt 5232  cfv 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557
This theorem is referenced by:  trlsfval  29601  pthsfval  29627  spthsfval  29628  clwlks  29678  crcts  29694  cycls  29695  eupths  30102
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