Theorem brfvopabrbr 6938

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 7413. (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 5148	. . . 4 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝐴‘𝑍) ≠ ∅)
2		fvprc 6826	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐴‘𝑍) = ∅)
3	2	necon1ai 2959	. . . 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 5769	. . . 4 ⊢ Rel (𝐴‘𝑍)
7	6	brrelex1i 5680	. . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑋 ∈ V)
8	6	brrelex2i 5681	. . 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 5148	. . . . 5 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝐵‘𝑍) ≠ ∅)
11		fvprc 6826	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐵‘𝑍) = ∅)
12	11	necon1ai 2959	. . . . 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 5680	. . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑋 ∈ V)
16	14	brrelex2i 5681	. . . 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 5511	. 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 1541   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440  ∅c0 4285   class class class wbr 5098  {copab 5160  Rel wrel 5629  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-iota 6448  df-fv 6500

This theorem is referenced by:  istrl  29768  ispth  29794  isspth  29795  isclwlk  29846  iscrct  29863  iscycl  29864  iseupth  30276