MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elovimad Structured version   Visualization version   GIF version

Theorem elovimad 7188
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 7143 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 elovimad.1 . . . 4 (𝜑𝐴𝐶)
3 elovimad.2 . . . 4 (𝜑𝐵𝐷)
42, 3opelxpd 5570 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))
5 elovimad.3 . . . 4 (𝜑 → Fun 𝐹)
6 elovimad.4 . . . . 5 (𝜑 → (𝐶 × 𝐷) ⊆ dom 𝐹)
76, 4sseldd 3943 . . . 4 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
8 funfvima 6975 . . . 4 ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ (𝐹 “ (𝐶 × 𝐷))))
95, 7, 8syl2anc 587 . . 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 2918 1 (𝜑 → (𝐴𝐹𝐵) ∈ (𝐹 “ (𝐶 × 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  wss 3908  cop 4545   × cxp 5530  dom cdm 5532  cima 5535  Fun wfun 6328  cfv 6334  (class class class)co 7140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342  df-ov 7143
This theorem is referenced by:  ltgov  26389  xrofsup  30502  icoreelrn  34739
  Copyright terms: Public domain W3C validator