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Theorem ovelrn 5570
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 𝐹x A y B 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 5567 . . 3 (𝐹 Fn (A × B) → ran 𝐹 = {zx A y B z = (x𝐹y)})
21eleq2d 2090 . 2 (𝐹 Fn (A × B) → (𝐶 ran 𝐹𝐶 {zx A y B z = (x𝐹y)}))
3 elex 2542 . . . 4 (𝐶 {zx A y B z = (x𝐹y)} → 𝐶 V)
43a1i 9 . . 3 (𝐹 Fn (A × B) → (𝐶 {zx A y B z = (x𝐹y)} → 𝐶 V))
5 fnovex 5460 . . . . . 6 ((𝐹 Fn (A × B) x A y B) → (x𝐹y) V)
6 eleq1 2083 . . . . . 6 (𝐶 = (x𝐹y) → (𝐶 V ↔ (x𝐹y) V))
75, 6syl5ibrcom 146 . . . . 5 ((𝐹 Fn (A × B) x A y B) → (𝐶 = (x𝐹y) → 𝐶 V))
873expb 1091 . . . 4 ((𝐹 Fn (A × B) (x A y B)) → (𝐶 = (x𝐹y) → 𝐶 V))
98rexlimdvva 2417 . . 3 (𝐹 Fn (A × B) → (x A y B 𝐶 = (x𝐹y) → 𝐶 V))
10 eqeq1 2029 . . . . . 6 (z = 𝐶 → (z = (x𝐹y) ↔ 𝐶 = (x𝐹y)))
11102rexbidv 2326 . . . . 5 (z = 𝐶 → (x A y B z = (x𝐹y) ↔ x A y B 𝐶 = (x𝐹y)))
1211elabg 2664 . . . 4 (𝐶 V → (𝐶 {zx A y B z = (x𝐹y)} ↔ x A y B 𝐶 = (x𝐹y)))
1312a1i 9 . . 3 (𝐹 Fn (A × B) → (𝐶 V → (𝐶 {zx A y B z = (x𝐹y)} ↔ x A y B 𝐶 = (x𝐹y))))
144, 9, 13pm5.21ndd 608 . 2 (𝐹 Fn (A × B) → (𝐶 {zx A y B z = (x𝐹y)} ↔ x A y B 𝐶 = (x𝐹y)))
152, 14bitrd 177 1 (𝐹 Fn (A × B) → (𝐶 ran 𝐹x A y B 𝐶 = (x𝐹y)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   w3a 873   = wceq 1228   wcel 1375  {cab 2009  wrex 2284  Vcvv 2534   × cxp 4268  ran crn 4271   Fn wfn 4822  (class class class)co 5434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-sbc 2741  df-csb 2829  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-iun 3632  df-br 3738  df-opab 3792  df-mpt 3793  df-id 4003  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-iota 4792  df-fun 4829  df-fn 4830  df-fv 4835  df-ov 5437
This theorem is referenced by: (None)
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