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Theorem ovelrn 6975
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 6972 . . 3 (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)})
21eleq2d 2825 . 2 (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)}))
3 ovex 6841 . . . . . 6 (𝑥𝐹𝑦) ∈ V
4 eleq1 2827 . . . . . 6 (𝐶 = (𝑥𝐹𝑦) → (𝐶 ∈ V ↔ (𝑥𝐹𝑦) ∈ V))
53, 4mpbiri 248 . . . . 5 (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V)
65rexlimivw 3167 . . . 4 (∃𝑦𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V)
76rexlimivw 3167 . . 3 (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V)
8 eqeq1 2764 . . . 4 (𝑧 = 𝐶 → (𝑧 = (𝑥𝐹𝑦) ↔ 𝐶 = (𝑥𝐹𝑦)))
982rexbidv 3195 . . 3 (𝑧 = 𝐶 → (∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))
107, 9elab3 3498 . 2 (𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)} ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦))
112, 10syl6bb 276 1 (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  Vcvv 3340   × cxp 5264  ran crn 5267   Fn wfn 6044  (class class class)co 6813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-ov 6816
This theorem is referenced by:  efgredlem  18360  efgcpbllemb  18368  gsumval3  18508  lecldbas  21225  blrnps  22414  blrn  22415  qdensere  22774  tgioo  22800  xrge0tsms  22838  ioorf  23541  ioorinv  23544  ioorcl  23545  dyaddisj  23564  dyadmax  23566  mbfid  23602  ismbfd  23606  hhssnv  28430  xrge0tsmsd  30094  iccllysconn  31539  rellysconn  31540  icoreelrnab  33513  relowlssretop  33522  relowlpssretop  33523  islptre  40354
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