Theorem opfv 32425

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) → (𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩)

Proof of Theorem opfv

Step	Hyp	Ref	Expression
1		simplr 768	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ran 𝐹 ⊆ (V × V))
2		fvelrn 7081	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
3	2	adantlr 714	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
4	1, 3	sseldd 3980	. . 3 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ (V × V))
5		1st2ndb 8028	. . 3 ⊢ ((𝐹‘𝑥) ∈ (V × V) ↔ (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
6	4, 5	sylib 217	. 2 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
7		fvco 6991	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥)))
8		fvco 6991	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥)))
9	7, 8	opeq12d 4878	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
10	9	adantlr 714	. 2 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
11	6, 10	eqtr4d 2771	1 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3470   ⊆ wss 3945  ⟨cop 4631   × cxp 5671  dom cdm 5673  ran crn 5674   ∘ ccom 5677  Fun wfun 6537  ‘cfv 6543  1^st c1st 7986  2^nd c2nd 7987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-1st 7988  df-2nd 7989

This theorem is referenced by:  xppreima  32426  xppreima2  32431