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Theorem ovelrn 7531
Description: A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
Assertion
Ref Expression
ovelrn (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦

Proof of Theorem ovelrn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 fnrnov 7528 . . 3 (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)})
21eleq2d 2820 . 2 (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)}))
3 ovex 7391 . . . . . 6 (𝑥𝐹𝑦) ∈ V
4 eleq1 2822 . . . . . 6 (𝐶 = (𝑥𝐹𝑦) → (𝐶 ∈ V ↔ (𝑥𝐹𝑦) ∈ V))
53, 4mpbiri 258 . . . . 5 (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V)
65rexlimivw 3145 . . . 4 (∃𝑦𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V)
76rexlimivw 3145 . . 3 (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V)
8 eqeq1 2737 . . . 4 (𝑧 = 𝐶 → (𝑧 = (𝑥𝐹𝑦) ↔ 𝐶 = (𝑥𝐹𝑦)))
982rexbidv 3210 . . 3 (𝑧 = 𝐶 → (∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))
107, 9elab3 3639 . 2 (𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)} ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦))
112, 10bitrdi 287 1 (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  {cab 2710  wrex 3070  Vcvv 3444   × cxp 5632  ran crn 5635   Fn wfn 6492  (class class class)co 7358
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 5257  ax-nul 5264  ax-pr 5385
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 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505  df-ov 7361
This theorem is referenced by:  efgredlem  19534  efgcpbllemb  19542  gsumval3  19689  lecldbas  22586  blrnps  23777  blrn  23778  qdensere  24149  tgioo  24175  xrge0tsms  24213  ioorf  24953  ioorinv  24956  ioorcl  24957  dyaddisj  24976  dyadmax  24978  mbfid  25015  ismbfd  25019  hhssnv  30248  xrge0tsmsd  31948  iccllysconn  33901  rellysconn  33902  icoreelrnab  35871  relowlssretop  35880  relowlpssretop  35881  islptre  43946
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