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Theorem opfv 32342
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 766 . . . 4 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ran 𝐹 ⊆ (V × V))
2 fvelrn 7069 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
32adantlr 712 . . . 4 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
41, 3sseldd 3976 . . 3 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ (V × V))
5 1st2ndb 8009 . . 3 ((𝐹𝑥) ∈ (V × V) ↔ (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
64, 5sylib 217 . 2 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
7 fvco 6980 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
8 fvco 6980 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
97, 8opeq12d 4874 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
109adantlr 712 . 2 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
116, 10eqtr4d 2767 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 395   = wceq 1533  wcel 2098  Vcvv 3466  wss 3941  cop 4627   × cxp 5665  dom cdm 5667  ran crn 5668  ccom 5671  Fun wfun 6528  cfv 6534  1st c1st 7967  2nd c2nd 7968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-fv 6542  df-1st 7969  df-2nd 7970
This theorem is referenced by:  xppreima  32343  xppreima2  32348
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