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Theorem fvconstr 48487
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 5170 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 fvconstr.1 . . . . . . 7 (𝜑𝐹 = (𝑅 × {𝑌}))
32oveqd 7462 . . . . . 6 (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑅 × {𝑌})𝐵))
4 df-ov 7448 . . . . . 6 (𝐴(𝑅 × {𝑌})𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩)
53, 4eqtrdi 2790 . . . . 5 (𝜑 → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩))
65adantr 480 . . . 4 ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴𝐹𝐵) = ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩))
7 fvconstr.2 . . . . 5 (𝜑𝑌𝑉)
8 fvconst2g 7237 . . . . 5 ((𝑌𝑉 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = 𝑌)
97, 8sylan 579 . . . 4 ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = 𝑌)
106, 9eqtrd 2774 . . 3 ((𝜑 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴𝐹𝐵) = 𝑌)
11 simpr 484 . . . . . 6 ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) = 𝑌)
12 fvconstr.3 . . . . . . 7 (𝜑𝑌 ≠ ∅)
1312adantr 480 . . . . . 6 ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → 𝑌 ≠ ∅)
1411, 13eqnetrd 3010 . . . . 5 ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → (𝐴𝐹𝐵) ≠ ∅)
155neeq1d 3002 . . . . . 6 (𝜑 → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅))
1615adantr 480 . . . . 5 ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ((𝐴𝐹𝐵) ≠ ∅ ↔ ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅))
1714, 16mpbid 232 . . . 4 ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅)
18 dmxpss 6201 . . . . 5 dom (𝑅 × {𝑌}) ⊆ 𝑅
19 ndmfv 6954 . . . . . 6 (¬ ⟨𝐴, 𝐵⟩ ∈ dom (𝑅 × {𝑌}) → ((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) = ∅)
2019necon1ai 2970 . . . . 5 (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom (𝑅 × {𝑌}))
2118, 20sselid 4000 . . . 4 (((𝑅 × {𝑌})‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2217, 21syl 17 . . 3 ((𝜑 ∧ (𝐴𝐹𝐵) = 𝑌) → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2310, 22impbida 800 . 2 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ (𝐴𝐹𝐵) = 𝑌))
241, 23bitrid 283 1 (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2103  wne 2942  c0 4347  {csn 4648  cop 4654   class class class wbr 5169   × cxp 5697  dom cdm 5699  cfv 6572  (class class class)co 7445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-fv 6580  df-ov 7448
This theorem is referenced by:  prsthinc  48639  prstchom2ALT  48664
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