Theorem brfvopabrbr 7026

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 7504. (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 5216	. . . 4 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝐴‘𝑍) ≠ ∅)
2		fvprc 6912	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐴‘𝑍) = ∅)
3	2	necon1ai 2974	. . . 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 5844	. . . 4 ⊢ Rel (𝐴‘𝑍)
7	6	brrelex1i 5756	. . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑋 ∈ V)
8	6	brrelex2i 5757	. . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑌 ∈ V)
9	4, 7, 8	3jca 1128	. 2 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
10		brne0 5216	. . . . 5 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝐵‘𝑍) ≠ ∅)
11		fvprc 6912	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐵‘𝑍) = ∅)
12	11	necon1ai 2974	. . . . 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 5756	. . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑋 ∈ V)
16	14	brrelex2i 5757	. . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑌 ∈ V)
17	13, 15, 16	3jca 1128	. . 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 5584	. 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 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488  ∅c0 4352   class class class wbr 5166  {copab 5228  Rel wrel 5705  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-iota 6525  df-fv 6581

This theorem is referenced by:  istrl  29732  ispth  29759  isspth  29760  isclwlk  29809  iscrct  29826  iscycl  29827  iseupth  30233