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Theorem ovelrn 5926
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 5923 . . 3 (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)})
21eleq2d 2210 . 2 (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)}))
3 elex 2700 . . . 4 (𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)} → 𝐶 ∈ V)
43a1i 9 . . 3 (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)} → 𝐶 ∈ V))
5 fnovex 5811 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥𝐴𝑦𝐵) → (𝑥𝐹𝑦) ∈ V)
6 eleq1 2203 . . . . . 6 (𝐶 = (𝑥𝐹𝑦) → (𝐶 ∈ V ↔ (𝑥𝐹𝑦) ∈ V))
75, 6syl5ibrcom 156 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥𝐴𝑦𝐵) → (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V))
873expb 1183 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥𝐴𝑦𝐵)) → (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V))
98rexlimdvva 2560 . . 3 (𝐹 Fn (𝐴 × 𝐵) → (∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V))
10 eqeq1 2147 . . . . . 6 (𝑧 = 𝐶 → (𝑧 = (𝑥𝐹𝑦) ↔ 𝐶 = (𝑥𝐹𝑦)))
11102rexbidv 2463 . . . . 5 (𝑧 = 𝐶 → (∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦) ↔ ∃𝑥𝐴𝑦𝐵 𝐶 = (𝑥𝐹𝑦)))
1211elabg 2833 . . . 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 695 . 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 963   = wceq 1332  wcel 1481  {cab 2126  wrex 2418  Vcvv 2689   × cxp 4544  ran crn 4547   Fn wfn 5125  (class class class)co 5781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-fv 5138  df-ov 5784
This theorem is referenced by:  blrnps  12617  blrn  12618  tgioo  12752
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