Theorem brfvopabrbr 6759

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 7203. (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 5108	. . . 4 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝐴‘𝑍) ≠ ∅)
2		fvprc 6657	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐴‘𝑍) = ∅)
3	2	necon1ai 3043	. . . 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 5688	. . . 4 ⊢ Rel (𝐴‘𝑍)
7	6	brrelex1i 5602	. . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑋 ∈ V)
8	6	brrelex2i 5603	. . 3 ⊢ (𝑋(𝐴‘𝑍)𝑌 → 𝑌 ∈ V)
9	4, 7, 8	3jca 1124	. 2 ⊢ (𝑋(𝐴‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
10		brne0 5108	. . . . 5 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝐵‘𝑍) ≠ ∅)
11		fvprc 6657	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → (𝐵‘𝑍) = ∅)
12	11	necon1ai 3043	. . . . 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 5602	. . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑋 ∈ V)
16	14	brrelex2i 5603	. . . 4 ⊢ (𝑋(𝐵‘𝑍)𝑌 → 𝑌 ∈ V)
17	13, 15, 16	3jca 1124	. . 3 ⊢ (𝑋(𝐵‘𝑍)𝑌 → (𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V))
18	17	adantr 483	. 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 5442	. 2 ⊢ ((𝑍 ∈ V ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓)))
22	9, 18, 21	pm5.21nii 382	1 ⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  Vcvv 3494  ∅c0 4290   class class class wbr 5058  {copab 5120  Rel wrel 5554  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-iota 6308  df-fv 6357

This theorem is referenced by:  istrl  27472  ispth  27498  isspth  27499  isclwlk  27548  iscrct  27565  iscycl  27566  iseupth  27974