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

Theorem ovelimab 7536
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 6918 . 2 ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑧 ∈ (𝐵 × 𝐶)(𝐹𝑧) = 𝐷))
2 fveq2 6846 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 7364 . . . . . 6 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3eqtr4di 2791 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝑥𝐹𝑦))
54eqeq1d 2735 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑧) = 𝐷 ↔ (𝑥𝐹𝑦) = 𝐷))
6 eqcom 2740 . . . 4 ((𝑥𝐹𝑦) = 𝐷𝐷 = (𝑥𝐹𝑦))
75, 6bitrdi 287 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑧) = 𝐷𝐷 = (𝑥𝐹𝑦)))
87rexxp 5802 . 2 (∃𝑧 ∈ (𝐵 × 𝐶)(𝐹𝑧) = 𝐷 ↔ ∃𝑥𝐵𝑦𝐶 𝐷 = (𝑥𝐹𝑦))
91, 8bitrdi 287 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 205  wa 397   = wceq 1542  wcel 2107  wrex 3070  wss 3914  cop 4596   × cxp 5635  cima 5640   Fn wfn 6495  cfv 6500  (class class class)co 7361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-ov 7364
This theorem is referenced by:  imaeqexov  7596  imaeqalov  7597  dfz2  12526  elq  12883  shsel  30305  ofrn2  31609  eulerpartlemgh  33042
  Copyright terms: Public domain W3C validator