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Theorem funfv1st2nd 7974
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 6515 . . 3 (Fun 𝐹 → Rel 𝐹)
2 1st2nd 7967 . . 3 ((Rel 𝐹𝑋𝐹) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
31, 2sylan 580 . 2 ((Fun 𝐹𝑋𝐹) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
4 eleq1 2825 . . . . 5 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋𝐹 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹))
54adantl 482 . . . 4 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐹 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹))
6 funopfv 6891 . . . . 5 (Fun 𝐹 → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
76adantr 481 . . . 4 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
85, 7sylbid 239 . . 3 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
98impancom 452 . 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 205  wa 396   = wceq 1541  wcel 2106  cop 4590  Rel wrel 5636  Fun wfun 6487  cfv 6493  1st c1st 7915  2nd c2nd 7916
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6445  df-fun 6495  df-fv 6501  df-1st 7917  df-2nd 7918
This theorem is referenced by:  gsumhashmul  31781  satffunlem  33864  satffunlem1lem1  33865  satffunlem2lem1  33867
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