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Theorem ovelrn 5831
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 5828 . . 3 (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)})
21eleq2d 2164 . 2 (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)}))
3 elex 2644 . . . 4 (𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)} → 𝐶 ∈ V)
43a1i 9 . . 3 (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)} → 𝐶 ∈ V))
5 fnovex 5720 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥𝐴𝑦𝐵) → (𝑥𝐹𝑦) ∈ V)
6 eleq1 2157 . . . . . 6 (𝐶 = (𝑥𝐹𝑦) → (𝐶 ∈ V ↔ (𝑥𝐹𝑦) ∈ V))
75, 6syl5ibrcom 156 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥𝐴𝑦𝐵) → (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V))
873expb 1147 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥𝐴𝑦𝐵)) → (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V))
98rexlimdvva 2510 . . 3 (𝐹 Fn (𝐴 × 𝐵) → (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V))
10 eqeq1 2101 . . . . . 6 (𝑧 = 𝐶 → (𝑧 = (𝑥𝐹𝑦) ↔ 𝐶 = (𝑥𝐹𝑦)))
11102rexbidv 2414 . . . . 5 (𝑧 = 𝐶 → (∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))
1211elabg 2775 . . . 4 (𝐶 ∈ V → (𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)} ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))
1312a1i 9 . . 3 (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ V → (𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)} ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦))))
144, 9, 13pm5.21ndd 659 . 2 (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)} ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))
152, 14bitrd 187 1 (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  w3a 927   = wceq 1296  wcel 1445  {cab 2081  wrex 2371  Vcvv 2633   × cxp 4465  ran crn 4468   Fn wfn 5044  (class class class)co 5690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fn 5052  df-fv 5057  df-ov 5693
This theorem is referenced by:  blrnps  12213  blrn  12214  tgioo  12336
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