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Theorem elovimad 6678
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 6638 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 elovimad.1 . . . 4 (𝜑𝐴𝐶)
3 elovimad.2 . . . 4 (𝜑𝐵𝐷)
4 opelxpi 5138 . . . 4 ((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
52, 3, 4syl2anc 692 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
6 elovimad.3 . . . 4 (𝜑 → Fun 𝐹)
7 elovimad.4 . . . . 5 (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹)
87, 5sseldd 3596 . . . 4 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
9 funfvima 6477 . . . 4 ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
106, 8, 9syl2anc 692 . . 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 2703 1 (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988  wss 3567  cop 4174   × cxp 5102  dom cdm 5104  cima 5107  Fun wfun 5870  cfv 5876  (class class class)co 6635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884  df-ov 6638
This theorem is referenced by:  ltgov  25473  xrofsup  29507  icoreelrn  33180
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