Theorem opfv 32705

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 769	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ran 𝐹 ⊆ (V × V))
2		fvelrn 7017	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
3	2	adantlr 716	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
4	1, 3	sseldd 3918	. . 3 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ (V × V))
5		1st2ndb 7971	. . 3 ⊢ ((𝐹‘𝑥) ∈ (V × V) ↔ (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
6	4, 5	sylib 218	. 2 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
7		fvco 6927	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥)))
8		fvco 6927	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥)))
9	7, 8	opeq12d 4814	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
10	9	adantlr 716	. 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 2773	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 1542   ∈ wcel 2114  Vcvv 3427   ⊆ wss 3885  ⟨cop 4563   × cxp 5618  dom cdm 5620  ran crn 5621   ∘ ccom 5624  Fun wfun 6481  ‘cfv 6487  1^st c1st 7929  2^nd c2nd 7930

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-fv 6495  df-1st 7931  df-2nd 7932

This theorem is referenced by:  xppreima  32706  xppreima2  32712