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Theorem opfv 29998
 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 787 . . . 4 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ran 𝐹 ⊆ (V × V))
2 fvelrn 6602 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
32adantlr 708 . . . 4 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
41, 3sseldd 3829 . . 3 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ (V × V))
5 1st2ndb 7469 . . 3 ((𝐹𝑥) ∈ (V × V) ↔ (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
64, 5sylib 210 . 2 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
7 fvco 6522 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
8 fvco 6522 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
97, 8opeq12d 4632 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
109adantlr 708 . 2 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
116, 10eqtr4d 2865 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 386   = wceq 1658   ∈ wcel 2166  Vcvv 3415   ⊆ wss 3799  ⟨cop 4404   × cxp 5341  dom cdm 5343  ran crn 5344   ∘ ccom 5347  Fun wfun 6118  ‘cfv 6124  1st c1st 7427  2nd c2nd 7428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-fv 6132  df-1st 7429  df-2nd 7430 This theorem is referenced by:  xppreima  29999  xppreima2  30000
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