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Theorem brfvopabrbr 6266
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 6682. (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 4693 . . . 4 (𝑋(𝐴𝑍)𝑌 → (𝐴𝑍) ≠ ∅)
2 fvprc 6172 . . . . 5 𝑍 ∈ V → (𝐴𝑍) = ∅)
32necon1ai 2818 . . . 4 ((𝐴𝑍) ≠ ∅ → 𝑍 ∈ V)
41, 3syl 17 . . 3 (𝑋(𝐴𝑍)𝑌𝑍 ∈ V)
5 brfvopabrbr.1 . . . . 5 (𝐴𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵𝑍)𝑦𝜑)}
65relopabi 5234 . . . 4 Rel (𝐴𝑍)
76brrelexi 5148 . . 3 (𝑋(𝐴𝑍)𝑌𝑋 ∈ V)
86brrelex2i 5149 . . 3 (𝑋(𝐴𝑍)𝑌𝑌 ∈ V)
94, 7, 83jca 1240 . 2 (𝑋(𝐴𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
10 brne0 4693 . . . . 5 (𝑋(𝐵𝑍)𝑌 → (𝐵𝑍) ≠ ∅)
11 fvprc 6172 . . . . . 6 𝑍 ∈ V → (𝐵𝑍) = ∅)
1211necon1ai 2818 . . . . 5 ((𝐵𝑍) ≠ ∅ → 𝑍 ∈ V)
1310, 12syl 17 . . . 4 (𝑋(𝐵𝑍)𝑌𝑍 ∈ V)
14 brfvopabrbr.3 . . . . 5 Rel (𝐵𝑍)
1514brrelexi 5148 . . . 4 (𝑋(𝐵𝑍)𝑌𝑋 ∈ V)
1614brrelex2i 5149 . . . 4 (𝑋(𝐵𝑍)𝑌𝑌 ∈ V)
1713, 15, 163jca 1240 . . 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 5006 . 2 ((𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋(𝐴𝑍)𝑌 ↔ (𝑋(𝐵𝑍)𝑌𝜓)))
229, 18, 21pm5.21nii 368 1 (𝑋(𝐴𝑍)𝑌 ↔ (𝑋(𝐵𝑍)𝑌𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  Vcvv 3195  c0 3907   class class class wbr 4644  {copab 4703  Rel wrel 5109  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-iota 5839  df-fv 5884
This theorem is referenced by:  istrl  26574  ispth  26600  isspth  26601  isclwlk  26650  iscrct  26666  iscycl  26667  iseupth  27041
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