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Theorem brfvopabrbr 6758
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 7198. (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 5107 . . . 4 (𝑋(𝐴𝑍)𝑌 → (𝐴𝑍) ≠ ∅)
2 fvprc 6656 . . . . 5 𝑍 ∈ V → (𝐴𝑍) = ∅)
32necon1ai 3040 . . . 4 ((𝐴𝑍) ≠ ∅ → 𝑍 ∈ V)
41, 3syl 17 . . 3 (𝑋(𝐴𝑍)𝑌𝑍 ∈ V)
5 brfvopabrbr.1 . . . . 5 (𝐴𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵𝑍)𝑦𝜑)}
65relopabi 5687 . . . 4 Rel (𝐴𝑍)
76brrelex1i 5601 . . 3 (𝑋(𝐴𝑍)𝑌𝑋 ∈ V)
86brrelex2i 5602 . . 3 (𝑋(𝐴𝑍)𝑌𝑌 ∈ V)
94, 7, 83jca 1120 . 2 (𝑋(𝐴𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
10 brne0 5107 . . . . 5 (𝑋(𝐵𝑍)𝑌 → (𝐵𝑍) ≠ ∅)
11 fvprc 6656 . . . . . 6 𝑍 ∈ V → (𝐵𝑍) = ∅)
1211necon1ai 3040 . . . . 5 ((𝐵𝑍) ≠ ∅ → 𝑍 ∈ V)
1310, 12syl 17 . . . 4 (𝑋(𝐵𝑍)𝑌𝑍 ∈ V)
14 brfvopabrbr.3 . . . . 5 Rel (𝐵𝑍)
1514brrelex1i 5601 . . . 4 (𝑋(𝐵𝑍)𝑌𝑋 ∈ V)
1614brrelex2i 5602 . . . 4 (𝑋(𝐵𝑍)𝑌𝑌 ∈ V)
1713, 15, 163jca 1120 . . 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 5441 . 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 1079   = wceq 1528  wcel 2105  wne 3013  Vcvv 3492  c0 4288   class class class wbr 5057  {copab 5119  Rel wrel 5553  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-iota 6307  df-fv 6356
This theorem is referenced by:  istrl  27405  ispth  27431  isspth  27432  isclwlk  27481  iscrct  27498  iscycl  27499  iseupth  27907
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