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Theorem funfv1st2nd 7979
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 6519 . . 3 (Fun 𝐹 → Rel 𝐹)
2 1st2nd 7972 . . 3 ((Rel 𝐹𝑋𝐹) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
31, 2sylan 581 . 2 ((Fun 𝐹𝑋𝐹) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
4 eleq1 2826 . . . . 5 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋𝐹 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹))
54adantl 483 . . . 4 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐹 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹))
6 funopfv 6895 . . . . 5 (Fun 𝐹 → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
76adantr 482 . . . 4 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
85, 7sylbid 239 . . 3 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
98impancom 453 . 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 397   = wceq 1542  wcel 2107  cop 4593  Rel wrel 5639  Fun wfun 6491  cfv 6497  1st c1st 7920  2nd c2nd 7921
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  gsumhashmul  31901  satffunlem  33998  satffunlem1lem1  33999  satffunlem2lem1  34001
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