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Theorem elovimad 7323
Description: Elementhood of the image set of an operation value. (Contributed by Thierry Arnoux, 13-Mar-2017.)
Hypotheses
Ref Expression
elovimad.1 (𝜑𝐴𝐶)
elovimad.2 (𝜑𝐵𝐷)
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
StepHypRef Expression
1 df-ov 7278 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 elovimad.1 . . . 4 (𝜑𝐴𝐶)
3 elovimad.2 . . . 4 (𝜑𝐵𝐷)
42, 3opelxpd 5627 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
5 elovimad.3 . . . 4 (𝜑 → Fun 𝐹)
6 elovimad.4 . . . . 5 (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹)
76, 4sseldd 3922 . . . 4 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
8 funfvima 7106 . . . 4 ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
95, 7, 8syl2anc 584 . . 3 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
104, 9mpd 15 . 2 (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷)))
111, 10eqeltrid 2843 1 (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wss 3887  cop 4567   × cxp 5587  dom cdm 5589  cima 5592  Fun wfun 6427  cfv 6433  (class class class)co 7275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-ov 7278
This theorem is referenced by:  ltgov  26958  xrofsup  31090  icoreelrn  35532
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