Theorem fvconstr 46183

Description: Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)

Hypotheses

Ref	Expression
fvconstr.1	⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))
fvconstr.2	⊢ (𝜑 → 𝑌 ∈ 𝑉)
fvconstr.3	⊢ (𝜑 → 𝑌 ≠ ∅)

Assertion

Ref	Expression
fvconstr	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌))

Proof of Theorem fvconstr

Step	Hyp	Ref	Expression
1		df-br 5075	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		fvconstr.1	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))
3	2	oveqd 7292	. . . . . 6 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵))
4		df-ov 7278	. . . . . 6 ⊢ (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩)
5	3, 4	eqtrdi 2794	. . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩))
6	5	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩))
7		fvconstr.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
8		fvconst2g 7077	. . . . 5 ⊢ ((𝑌 ∈ 𝑉 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = 𝑌)
9	7, 8	sylan 580	. . . 4 ⊢ ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = 𝑌)
10	6, 9	eqtrd 2778	. . 3 ⊢ ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴𝐹𝐵) = 𝑌)
11		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) = 𝑌)
12		fvconstr.3	. . . . . . 7 ⊢ (𝜑 → 𝑌 ≠ ∅)
13	12	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → 𝑌 ≠ ∅)
14	11, 13	eqnetrd 3011	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) ≠ ∅)
15	5	neeq1d 3003	. . . . . 6 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅))
16	15	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅))
17	14, 16	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅)
18		dmxpss 6074	. . . . 5 ⊢ dom (𝑅 × {𝑌}) ⊆ 𝑅
19		ndmfv 6804	. . . . . 6 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = ∅)
20	19	necon1ai 2971	. . . . 5 ⊢ (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom (𝑅 × {𝑌}))
21	18, 20	sselid 3919	. . . 4 ⊢ (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
22	17, 21	syl 17	. . 3 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
23	10, 22	impbida 798	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ (𝐴𝐹𝐵) = 𝑌))
24	1, 23	syl5bb 283	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256  {csn 4561  ⟨cop 4567   class class class wbr 5074   × cxp 5587  dom cdm 5589  ‘cfv 6433  (class class class)co 7275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278

This theorem is referenced by:  prsthinc  46335  prstchom2ALT  46360