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Theorem brfvopabrbr 6421
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 6844. (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 4836 . . . 4 (𝑋(𝐴𝑍)𝑌 → (𝐴𝑍) ≠ ∅)
2 fvprc 6326 . . . . 5 𝑍 ∈ V → (𝐴𝑍) = ∅)
32necon1ai 2970 . . . 4 ((𝐴𝑍) ≠ ∅ → 𝑍 ∈ V)
41, 3syl 17 . . 3 (𝑋(𝐴𝑍)𝑌𝑍 ∈ V)
5 brfvopabrbr.1 . . . . 5 (𝐴𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵𝑍)𝑦𝜑)}
65relopabi 5384 . . . 4 Rel (𝐴𝑍)
76brrelexi 5298 . . 3 (𝑋(𝐴𝑍)𝑌𝑋 ∈ V)
86brrelex2i 5299 . . 3 (𝑋(𝐴𝑍)𝑌𝑌 ∈ V)
94, 7, 83jca 1122 . 2 (𝑋(𝐴𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
10 brne0 4836 . . . . 5 (𝑋(𝐵𝑍)𝑌 → (𝐵𝑍) ≠ ∅)
11 fvprc 6326 . . . . . 6 𝑍 ∈ V → (𝐵𝑍) = ∅)
1211necon1ai 2970 . . . . 5 ((𝐵𝑍) ≠ ∅ → 𝑍 ∈ V)
1310, 12syl 17 . . . 4 (𝑋(𝐵𝑍)𝑌𝑍 ∈ V)
14 brfvopabrbr.3 . . . . 5 Rel (𝐵𝑍)
1514brrelexi 5298 . . . 4 (𝑋(𝐵𝑍)𝑌𝑋 ∈ V)
1614brrelex2i 5299 . . . 4 (𝑋(𝐵𝑍)𝑌𝑌 ∈ V)
1713, 15, 163jca 1122 . . 3 (𝑋(𝐵𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
1817adantr 466 . 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 5149 . 2 ((𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋(𝐴𝑍)𝑌 ↔ (𝑋(𝐵𝑍)𝑌𝜓)))
229, 18, 21pm5.21nii 367 1 (𝑋(𝐴𝑍)𝑌 ↔ (𝑋(𝐵𝑍)𝑌𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  c0 4063   class class class wbr 4786  {copab 4846  Rel wrel 5254  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-iota 5994  df-fv 6039
This theorem is referenced by:  istrl  26828  ispth  26854  isspth  26855  isclwlk  26904  iscrct  26921  iscycl  26922  iseupth  27381
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