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Theorem funfv1st2nd 8071
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 6583 . . 3 (Fun 𝐹 → Rel 𝐹)
2 1st2nd 8064 . . 3 ((Rel 𝐹𝑋𝐹) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
31, 2sylan 580 . 2 ((Fun 𝐹𝑋𝐹) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
4 eleq1 2829 . . . . 5 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋𝐹 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹))
54adantl 481 . . . 4 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐹 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹))
6 funopfv 6958 . . . . 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 1540  wcel 2108  cop 4632  Rel wrel 5690  Fun wfun 6555  cfv 6561  1st c1st 8012  2nd c2nd 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015
This theorem is referenced by:  gsumhashmul  33064  satffunlem  35406  satffunlem1lem1  35407  satffunlem2lem1  35409
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