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Theorem funfv1st2nd 7749
 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 6352 . . 3 (Fun 𝐹 → Rel 𝐹)
2 1st2nd 7742 . . 3 ((Rel 𝐹𝑋𝐹) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
31, 2sylan 583 . 2 ((Fun 𝐹𝑋𝐹) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
4 eleq1 2839 . . . . 5 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋𝐹 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹))
54adantl 485 . . . 4 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐹 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹))
6 funopfv 6705 . . . . 5 (Fun 𝐹 → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
76adantr 484 . . . 4 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
85, 7sylbid 243 . . 3 ((Fun 𝐹𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐹 → (𝐹‘(1st𝑋)) = (2nd𝑋)))
98impancom 455 . 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4528  Rel wrel 5529  Fun wfun 6329  ‘cfv 6335  1st c1st 7691  2nd c2nd 7692 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-iota 6294  df-fun 6337  df-fv 6343  df-1st 7693  df-2nd 7694 This theorem is referenced by:  gsumhashmul  30842  satffunlem  32879  satffunlem1lem1  32880  satffunlem2lem1  32882
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