Theorem brfvopabrbr 7012

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 7486. (Contributed by AV, 29-Oct-2021.)

Hypotheses

Ref	Expression
brfvopabrbr.1	⊢ (𝐴‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵‘𝑍)𝑦 ∧ 𝜑)}
brfvopabrbr.2	⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))
brfvopabrbr.3	⊢ Rel (𝐵‘𝑍)

Assertion

Ref	Expression
brfvopabrbr	⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓))

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

Step	Hyp	Ref	Expression
1		brne0 5197	. . . 4 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝐴‘𝑍) ≠ ∅)
2		fvprc 6898	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐴‘𝑍) = ∅)
3	2	necon1ai 2965	. . . 4 ⊢ ((𝐴‘𝑍) ≠ ∅ → 𝑍 ∈ V)
4	1, 3	syl 17	. . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑍 ∈ V)
5		brfvopabrbr.1	. . . . 5 ⊢ (𝐴‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵‘𝑍)𝑦 ∧ 𝜑)}
6	5	relopabiv 5832	. . . 4 ⊢ Rel (𝐴‘𝑍)
7	6	brrelex1i 5744	. . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑋 ∈ V)
8	6	brrelex2i 5745	. . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑌 ∈ V)
9	4, 7, 8	3jca 1127	. 2 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
10		brne0 5197	. . . . 5 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝐵‘𝑍) ≠ ∅)
11		fvprc 6898	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐵‘𝑍) = ∅)
12	11	necon1ai 2965	. . . . 5 ⊢ ((𝐵‘𝑍) ≠ ∅ → 𝑍 ∈ V)
13	10, 12	syl 17	. . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑍 ∈ V)
14		brfvopabrbr.3	. . . . 5 ⊢ Rel (𝐵‘𝑍)
15	14	brrelex1i 5744	. . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑋 ∈ V)
16	14	brrelex2i 5745	. . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑌 ∈ V)
17	13, 15, 16	3jca 1127	. . 3 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
18	17	adantr 480	. 2 ⊢ ((𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓) → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
19	5	a1i 11	. . 3 ⊢ (𝑍 ∈ V → (𝐴‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵‘𝑍)𝑦 ∧ 𝜑)})
20		brfvopabrbr.2	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))
21	19, 20	rbropap 5574	. 2 ⊢ ((𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓)))
22	9, 18, 21	pm5.21nii 378	1 ⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  Vcvv 3477  ∅c0 4338   class class class wbr 5147  {copab 5209  Rel wrel 5693  ‘cfv 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-iota 6515  df-fv 6570

This theorem is referenced by:  istrl  29728  ispth  29755  isspth  29756  isclwlk  29805  iscrct  29822  iscycl  29823  iseupth  30229