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Theorem fvconstr 46183
Description: Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
Hypotheses
Ref Expression
fvconstr.1 (𝜑𝐹 = (𝑅 × {𝑌}))
fvconstr.2 (𝜑𝑌𝑉)
fvconstr.3 (𝜑𝑌 ≠ ∅)
Assertion
Ref Expression
fvconstr (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌))

Proof of Theorem fvconstr
StepHypRef Expression
1 df-br 5075 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 fvconstr.1 . . . . . . 7 (𝜑𝐹 = (𝑅 × {𝑌}))
32oveqd 7292 . . . . . 6 (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵))
4 df-ov 7278 . . . . . 6 (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩)
53, 4eqtrdi 2794 . . . . 5 (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩))
65adantr 481 . . . 4 ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩))
7 fvconstr.2 . . . . 5 (𝜑𝑌𝑉)
8 fvconst2g 7077 . . . . 5 ((𝑌𝑉 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = 𝑌)
97, 8sylan 580 . . . 4 ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = 𝑌)
106, 9eqtrd 2778 . . 3 ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴𝐹𝐵) = 𝑌)
11 simpr 485 . . . . . 6 ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) = 𝑌)
12 fvconstr.3 . . . . . . 7 (𝜑𝑌 ≠ ∅)
1312adantr 481 . . . . . 6 ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → 𝑌 ≠ ∅)
1411, 13eqnetrd 3011 . . . . 5 ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) ≠ ∅)
155neeq1d 3003 . . . . . 6 (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅))
1615adantr 481 . . . . 5 ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅))
1714, 16mpbid 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⟩) = ∅)
2019necon1ai 2971 . . . . 5 (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom (𝑅 × {𝑌}))
2118, 20sselid 3919 . . . 4 (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2217, 21syl 17 . . 3 ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2310, 22impbida 798 . 2 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ (𝐴𝐹𝐵) = 𝑌))
241, 23syl5bb 283 1 (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌))
Colors of variables: wff setvar class
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
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