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

Theorem ovelimab 7072
Description: Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
Assertion
Ref Expression
ovelimab ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑥𝐵𝑦𝐶 𝐷 = (𝑥𝐹𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐹,𝑦

Proof of Theorem ovelimab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 fvelimab 6500 . 2 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑧 ∈ (𝐵 × 𝐶)(𝐹𝑧) = 𝐷))
2 fveq2 6433 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 6908 . . . . . 6 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3syl6eqr 2879 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝑥𝐹𝑦))
54eqeq1d 2827 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑧) = 𝐷 ↔ (𝑥𝐹𝑦) = 𝐷))
6 eqcom 2832 . . . 4 ((𝑥𝐹𝑦) = 𝐷𝐷 = (𝑥𝐹𝑦))
75, 6syl6bb 279 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑧) = 𝐷𝐷 = (𝑥𝐹𝑦)))
87rexxp 5497 . 2 (∃𝑧 ∈ (𝐵 × 𝐶)(𝐹𝑧) = 𝐷 ↔ ∃𝑥𝐵𝑦𝐶 𝐷 = (𝑥𝐹𝑦))
91, 8syl6bb 279 1 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑥𝐵𝑦𝐶 𝐷 = (𝑥𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wrex 3118  wss 3798  cop 4403   × cxp 5340  cima 5345   Fn wfn 6118  cfv 6123  (class class class)co 6905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131  df-ov 6908
This theorem is referenced by:  dfz2  11722  elq  12073  shsel  28728  ofrn2  29991  eulerpartlemgh  30985
  Copyright terms: Public domain W3C validator