Theorem fvconstr2 49800

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

Hypotheses

Ref	Expression
fvconstr.1	⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))
fvconstr2.2	⊢ (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵))

Assertion

Ref	Expression
fvconstr2	⊢ (𝜑 → 𝐴𝑅𝐵)

Proof of Theorem fvconstr2

Step	Hyp	Ref	Expression
1		fvconstr2.2	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵))
2	1	ne0d 4291	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ≠ ∅)
3		fvconstr.1	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))
4	3	oveqd 7434	. . . . . 6 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵))
5		df-ov 7420	. . . . . 6 ⊢ (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩)
6	4, 5	eqtrdi 2813	. . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩))
7	6	neeq1d 3016	. . . 4 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅))
8		dmxpss 6168	. . . . 5 ⊢ dom (𝑅 × {𝑌}) ⊆ 𝑅
9		ndmfv 6914	. . . . . 6 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = ∅)
10	9	necon1ai 2984	. . . . 5 ⊢ (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom (𝑅 × {𝑌}))
11	8, 10	sselid 3932	. . . 4 ⊢ (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
12	7, 11	biimtrdi 256	. . 3 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑅))
13	2, 12	mpd 16	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
14		df-br 5108	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
15	13, 14	sylibr 237	1 ⊢ (𝜑 → 𝐴𝑅𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∅c0 4282  {csn 4587  ⟨cop 4593   class class class wbr 5107   × cxp 5657  dom cdm 5659  ‘cfv 6537  (class class class)co 7417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-dm 5669  df-iota 6493  df-fv 6545  df-ov 7420

This theorem is used by:  prsthinc  50398