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Theorem funfv1st2nd 8070
Description: The function value for the first component of an ordered pair is the second component of the ordered pair. (Contributed by AV, 17-Oct-2023.)
Assertion
Ref Expression
funfv1st2nd ((Fun 𝐹𝑋𝐹) → (𝐹‘(1st𝑋)) = (2nd𝑋))

Proof of Theorem funfv1st2nd
StepHypRef Expression
1 funrel 6585 . . 3 (Fun 𝐹 → Rel 𝐹)
2 1st2nd 8063 . . 3 ((Rel 𝐹𝑋𝐹) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
31, 2sylan 580 . 2 ((Fun 𝐹𝑋𝐹) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
4 eleq1 2827 . . . . 5 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋𝐹 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹))
54adantl 481 . . . 4 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐹 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹))
6 funopfv 6959 . . . . 5 (Fun 𝐹 → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
76adantr 480 . . . 4 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
85, 7sylbid 240 . . 3 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
98impancom 451 . 2 ((Fun 𝐹𝑋𝐹) → (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝐹‘(1st𝑋)) = (2nd𝑋)))
103, 9mpd 15 1 ((Fun 𝐹𝑋𝐹) → (𝐹‘(1st𝑋)) = (2nd𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cop 4637  Rel wrel 5694  Fun wfun 6557  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  gsumhashmul  33047  satffunlem  35386  satffunlem1lem1  35387  satffunlem2lem1  35389
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