Theorem fvconstr 49144

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 5098	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2		fvconstr.1	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))
3	2	oveqd 7375	. . . . . 6 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵))
4		df-ov 7361	. . . . . 6 ⊢ (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩)
5	3, 4	eqtrdi 2786	. . . . 5 ⊢ (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩))
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩))
7		fvconstr.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
8		fvconst2g 7148	. . . . 5 ⊢ ((𝑌 ∈ 𝑉 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = 𝑌)
9	7, 8	sylan 581	. . . 4 ⊢ ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = 𝑌)
10	6, 9	eqtrd 2770	. . 3 ⊢ ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴𝐹𝐵) = 𝑌)
11		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) = 𝑌)
12		fvconstr.3	. . . . . . 7 ⊢ (𝜑 → 𝑌 ≠ ∅)
13	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → 𝑌 ≠ ∅)
14	11, 13	eqnetrd 2998	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) ≠ ∅)
15	5	neeq1d 2990	. . . . . 6 ⊢ (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅))
16	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅))
17	14, 16	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅)
18		dmxpss 6128	. . . . 5 ⊢ dom (𝑅 × {𝑌}) ⊆ 𝑅
19		ndmfv 6865	. . . . . 6 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = ∅)
20	19	necon1ai 2958	. . . . 5 ⊢ (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom (𝑅 × {𝑌}))
21	18, 20	sselid 3930	. . . 4 ⊢ (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
22	17, 21	syl 17	. . 3 ⊢ ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
23	10, 22	impbida 801	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ (𝐴𝐹𝐵) = 𝑌))
24	1, 23	bitrid 283	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2931  ∅c0 4284  {csn 4579  ⟨cop 4585   class class class wbr 5097   × cxp 5621  dom cdm 5623  ‘cfv 6491  (class class class)co 7358

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-fv 6499  df-ov 7361

This theorem is referenced by:  prsthinc  49746  prstchom2ALT  49846