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 Description: Elementhood of the image set of an operation value. (Contributed by Thierry Arnoux, 13-Mar-2017.)
Hypotheses
Ref Expression
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 6552 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 elovimad.1 . . . 4 (𝜑𝐴𝐶)
3 elovimad.2 . . . 4 (𝜑𝐵𝐷)
4 opelxpi 5072 . . . 4 ((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
52, 3, 4syl2anc 691 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
6 elovimad.3 . . . 4 (𝜑 → Fun 𝐹)
7 elovimad.4 . . . . 5 (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹)
87, 5sseldd 3569 . . . 4 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
9 funfvima 6396 . . . 4 ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
106, 8, 9syl2anc 691 . . 3 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
115, 10mpd 15 . 2 (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷)))
121, 11syl5eqel 2692 1 (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ⊆ wss 3540  ⟨cop 4131   × cxp 5036  dom cdm 5038   “ cima 5041  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552 This theorem is referenced by:  ltgov  25292  xrofsup  28923  icoreelrn  32385
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