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Theorem brfvopabrbr 6764
Description: The binary relation of a function value which is an ordered-pair class abstraction of a restricted binary relation is the restricted binary relation. The first hypothesis can often be obtained by using fvmptopab 7203. (Contributed by AV, 29-Oct-2021.)
Hypotheses
Ref Expression
brfvopabrbr.1 (𝐴𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵𝑍)𝑦𝜑)}
brfvopabrbr.2 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))
brfvopabrbr.3 Rel (𝐵𝑍)
Assertion
Ref Expression
brfvopabrbr (𝑋(𝐴𝑍)𝑌 ↔ (𝑋(𝐵𝑍)𝑌𝜓))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑍,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem brfvopabrbr
StepHypRef Expression
1 brne0 5113 . . . 4 (𝑋(𝐴𝑍)𝑌 → (𝐴𝑍) ≠ ∅)
2 fvprc 6662 . . . . 5 𝑍 ∈ V → (𝐴𝑍) = ∅)
32necon1ai 3048 . . . 4 ((𝐴𝑍) ≠ ∅ → 𝑍 ∈ V)
41, 3syl 17 . . 3 (𝑋(𝐴𝑍)𝑌𝑍 ∈ V)
5 brfvopabrbr.1 . . . . 5 (𝐴𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵𝑍)𝑦𝜑)}
65relopabi 5693 . . . 4 Rel (𝐴𝑍)
76brrelex1i 5607 . . 3 (𝑋(𝐴𝑍)𝑌𝑋 ∈ V)
86brrelex2i 5608 . . 3 (𝑋(𝐴𝑍)𝑌𝑌 ∈ V)
94, 7, 83jca 1122 . 2 (𝑋(𝐴𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
10 brne0 5113 . . . . 5 (𝑋(𝐵𝑍)𝑌 → (𝐵𝑍) ≠ ∅)
11 fvprc 6662 . . . . . 6 𝑍 ∈ V → (𝐵𝑍) = ∅)
1211necon1ai 3048 . . . . 5 ((𝐵𝑍) ≠ ∅ → 𝑍 ∈ V)
1310, 12syl 17 . . . 4 (𝑋(𝐵𝑍)𝑌𝑍 ∈ V)
14 brfvopabrbr.3 . . . . 5 Rel (𝐵𝑍)
1514brrelex1i 5607 . . . 4 (𝑋(𝐵𝑍)𝑌𝑋 ∈ V)
1614brrelex2i 5608 . . . 4 (𝑋(𝐵𝑍)𝑌𝑌 ∈ V)
1713, 15, 163jca 1122 . . 3 (𝑋(𝐵𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
1817adantr 481 . 2 ((𝑋(𝐵𝑍)𝑌𝜓) → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
195a1i 11 . . 3 (𝑍 ∈ V → (𝐴𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵𝑍)𝑦𝜑)})
20 brfvopabrbr.2 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))
2119, 20rbropap 5447 . 2 ((𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋(𝐴𝑍)𝑌 ↔ (𝑋(𝐵𝑍)𝑌𝜓)))
229, 18, 21pm5.21nii 380 1 (𝑋(𝐴𝑍)𝑌 ↔ (𝑋(𝐵𝑍)𝑌𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  Vcvv 3500  c0 4295   class class class wbr 5063  {copab 5125  Rel wrel 5559  cfv 6354
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-xp 5560  df-rel 5561  df-iota 6313  df-fv 6362
This theorem is referenced by:  istrl  27411  ispth  27437  isspth  27438  isclwlk  27487  iscrct  27504  iscycl  27505  iseupth  27913
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