elovimad - Metamath Proof Explorer

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Theorem elovimad 7467

Description: Elementhood of the image set of an operation value. (Contributed by Thierry Arnoux, 13-Mar-2017.)

**Hypotheses**
Ref	Expression
elovimad.1	⊢ (𝜑 → 𝐴 ∈ 𝐶)
elovimad.2	⊢ (𝜑 → 𝐵 ∈ 𝐷)
elovimad.3	⊢ (𝜑 → Fun 𝐹)
elovimad.4	⊢ (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹)

**Assertion**
Ref	Expression
elovimad	⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))

**Proof of Theorem elovimad**
Step	Hyp	Ref	Expression
1		df-ov 7420	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		elovimad.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
3		elovimad.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
4	2, 3	opelxpd 5698	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
5		elovimad.3	. . . 4 ⊢ (𝜑 → Fun 𝐹)
6		elovimad.4	. . . . 5 ⊢ (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹)
7	6, 4	sseldd 3935	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
8		funfvima 7233	. . . 4 ⊢ ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
9	5, 7, 8	syl2anc 596	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
10	4, 9	mpd 16	. 2 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷)))
11	1, 10	eqeltrid 2866	1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2145 ⊆ wss 3902 ⟨cop 4593 × cxp 5657 dom cdm 5659 “ cima 5662 Fun wfun 6531 ‘cfv 6537 (class class class)co 7417

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-fv 6545 df-ov 7420

This theorem is used by: psdmul 22400 ltgov 28947 xrofsup 33246 icoreelrn 38123

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