opfv - Mathbox for Thierry Arnoux

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Theorem opfv 30393

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 767	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ran 𝐹 ⊆ (V × V))
2		fvelrn 6844	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
3	2	adantlr 713	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
4	1, 3	sseldd 3968	. . 3 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ (V × V))
5		1st2ndb 7729	. . 3 ⊢ ((𝐹‘𝑥) ∈ (V × V) ↔ (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
6	4, 5	sylib 220	. 2 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
7		fvco 6759	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((1^st ∘ 𝐹)‘𝑥) = (1^st ‘(𝐹‘𝑥)))
8		fvco 6759	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((2^nd ∘ 𝐹)‘𝑥) = (2^nd ‘(𝐹‘𝑥)))
9	7, 8	opeq12d 4811	. . 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 713	. 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 2859	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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ⟨cop 4573 × cxp 5553 dom cdm 5555 ran crn 5556 ∘ ccom 5559 Fun wfun 6349 ‘cfv 6355 1^st c1st 7687 2^nd c2nd 7688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 df-1st 7689 df-2nd 7690

This theorem is referenced by: xppreima 30394 xppreima2 30395

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