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Theorem elovimad 6842
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 6799 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 elovimad.1 . . . 4 (𝜑𝐴𝐶)
3 elovimad.2 . . . 4 (𝜑𝐵𝐷)
4 opelxpi 5287 . . . 4 ((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
52, 3, 4syl2anc 573 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
6 elovimad.3 . . . 4 (𝜑 → Fun 𝐹)
7 elovimad.4 . . . . 5 (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹)
87, 5sseldd 3753 . . . 4 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
9 funfvima 6638 . . . 4 ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
106, 8, 9syl2anc 573 . . 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 2854 1 (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  wss 3723  cop 4323   × cxp 5248  dom cdm 5250  cima 5253  Fun wfun 6024  cfv 6030  (class class class)co 6796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-fv 6038  df-ov 6799
This theorem is referenced by:  ltgov  25713  xrofsup  29873  icoreelrn  33546
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