Theorem fvconstr2 45801

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 4236	. . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ≠ ∅)
3		fvconstr.1	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))
4	3	oveqd 7208	. . . . . 6 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵))
5		df-ov 7194	. . . . . 6 ⊢ (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩)
6	4, 5	eqtrdi 2787	. . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩))
7	6	neeq1d 2991	. . . 4 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅))
8		dmxpss 6014	. . . . 5 ⊢ dom (𝑅 × {𝑌}) ⊆ 𝑅
9		ndmfv 6725	. . . . . 6 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = ∅)
10	9	necon1ai 2959	. . . . 5 ⊢ (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom (𝑅 × {𝑌}))
11	8, 10	sseldi 3885	. . . 4 ⊢ (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
12	7, 11	syl6bi 256	. . 3 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑅))
13	2, 12	mpd 15	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
14		df-br 5040	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
15	13, 14	sylibr 237	1 ⊢ (𝜑 → 𝐴𝑅𝐵)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2112   ≠ wne 2932  ∅c0 4223  {csn 4527  ⟨cop 4533   class class class wbr 5039   × cxp 5534  dom cdm 5536  ‘cfv 6358  (class class class)co 7191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-xp 5542  df-rel 5543  df-cnv 5544  df-dm 5546  df-iota 6316  df-fv 6366  df-ov 7194

This theorem is referenced by:  prsthinc  45951