Theorem brfvopabrbr 6742

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 7188. (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 5080	. . . 4 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝐴‘𝑍) ≠ ∅)
2		fvprc 6638	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐴‘𝑍) = ∅)
3	2	necon1ai 3014	. . . 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	relopabi 5658	. . . 4 ⊢ Rel (𝐴‘𝑍)
7	6	brrelex1i 5572	. . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑋 ∈ V)
8	6	brrelex2i 5573	. . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑌 ∈ V)
9	4, 7, 8	3jca 1125	. 2 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
10		brne0 5080	. . . . 5 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝐵‘𝑍) ≠ ∅)
11		fvprc 6638	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐵‘𝑍) = ∅)
12	11	necon1ai 3014	. . . . 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 5572	. . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑋 ∈ V)
16	14	brrelex2i 5573	. . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑌 ∈ V)
17	13, 15, 16	3jca 1125	. . 3 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
18	17	adantr 484	. 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 5415	. 2 ⊢ ((𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓)))
22	9, 18, 21	pm5.21nii 383	1 ⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441  ∅c0 4243   class class class wbr 5030  {copab 5092  Rel wrel 5524  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-iota 6283  df-fv 6332

This theorem is referenced by:  istrl  27486  ispth  27512  isspth  27513  isclwlk  27562  iscrct  27579  iscycl  27580  iseupth  27986