opfv - Mathbox for Thierry Arnoux

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Theorem opfv 30883

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) = ⟨((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)⟩)

**Proof of Theorem opfv**
Step	Hyp	Ref	Expression
1		simplr 765	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ran 𝐹 ⊆ (V × V))
2		fvelrn 6936	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
3	2	adantlr 711	. . . 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 7844	. . 3 ⊢ ((𝐹‘𝑥) ∈ (V × V) ↔ (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
6	4, 5	sylib 217	. 2 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
7		fvco 6848	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((1^st ∘ 𝐹)‘𝑥) = (1^st ‘(𝐹‘𝑥)))
8		fvco 6848	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((2^nd ∘ 𝐹)‘𝑥) = (2^nd ‘(𝐹‘𝑥)))
9	7, 8	opeq12d 4809	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)⟩ = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
10	9	adantlr 711	. 2 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ⟨((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)⟩ = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
11	6, 10	eqtr4d 2781	1 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ⊆ wss 3883 ⟨cop 4564 × cxp 5578 dom cdm 5580 ran crn 5581 ∘ ccom 5584 Fun wfun 6412 ‘cfv 6418 1^st c1st 7802 2^nd c2nd 7803

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-fv 6426 df-1st 7804 df-2nd 7805

This theorem is referenced by: xppreima 30884 xppreima2 30889

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