Theorem elovimad 7460

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 7415	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		elovimad.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
3		elovimad.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
4	2, 3	opelxpd 5715	. . 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 3983	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
8		funfvima 7234	. . . 4 ⊢ ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
9	5, 7, 8	syl2anc 583	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
10	4, 9	mpd 15	. 2 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷)))
11	1, 10	eqeltrid 2836	1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))

Syntax hints:   → wi 4   ∈ wcel 2105   ⊆ wss 3948  ⟨cop 4634   × cxp 5674  dom cdm 5676   “ cima 5679  Fun wfun 6537  ‘cfv 6543  (class class class)co 7412

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-ov 7415

This theorem is referenced by:  ltgov  28116  xrofsup  32248  icoreelrn  36546