Theorem elovimad 7408

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 7361	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		elovimad.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
3		elovimad.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
4	2, 3	opelxpd 5663	. . 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 3934	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
8		funfvima 7176	. . . 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 2840	1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))

Syntax hints:   → wi 4   ∈ wcel 2113   ⊆ wss 3901  ⟨cop 4586   × cxp 5622  dom cdm 5624   “ cima 5627  Fun wfun 6486  ‘cfv 6492  (class class class)co 7358

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-ov 7361

This theorem is referenced by:  psdmul  22109  ltgov  28669  xrofsup  32847  icoreelrn  37562