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Theorem opfv 28828
 Description: Value of a function producing ordered pairs. (Contributed by Thierry Arnoux, 3-Jan-2017.)
Assertion
Ref Expression
opfv (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩)

Proof of Theorem opfv
StepHypRef Expression
1 simplr 788 . . . 4 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ran 𝐹 ⊆ (V × V))
2 fvelrn 6260 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
32adantlr 747 . . . 4 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
41, 3sseldd 3569 . . 3 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ (V × V))
5 1st2ndb 7097 . . 3 ((𝐹𝑥) ∈ (V × V) ↔ (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
64, 5sylib 207 . 2 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
7 fvco 6184 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
8 fvco 6184 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
97, 8opeq12d 4348 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
109adantlr 747 . 2 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
116, 10eqtr4d 2647 1 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  Fun wfun 5798  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-1st 7059  df-2nd 7060 This theorem is referenced by:  xppreima  28829  xppreima2  28830
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