Theorem elovimad 7481

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 7434	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		elovimad.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
3		elovimad.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
4	2, 3	opelxpd 5728	. . 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 3996	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
8		funfvima 7250	. . . 4 ⊢ ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
9	5, 7, 8	syl2anc 584	. . 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 2843	1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))

Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3963  ⟨cop 4637   × cxp 5687  dom cdm 5689   “ cima 5692  Fun wfun 6557  ‘cfv 6563  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434

This theorem is referenced by:  psdmul  22188  ltgov  28620  xrofsup  32778  icoreelrn  37344