ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ovelrn Structured version   GIF version

Theorem ovelrn 5541
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 5538 . . 3 (𝐹 Fn (A × B) → ran 𝐹 = {zx A y B z = (x𝐹y)})
21eleq2d 2089 . 2 (𝐹 Fn (A × B) → (𝐶 ran 𝐹𝐶 {zx A y B z = (x𝐹y)}))
3 elex 2541 . . . 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 5431 . . . . . 6 ((𝐹 Fn (A × B) x A y B) → (x𝐹y) V)
6 eleq1 2082 . . . . . 6 (𝐶 = (x𝐹y) → (𝐶 V ↔ (x𝐹y) V))
75, 6syl5ibrcom 146 . . . . 5 ((𝐹 Fn (A × B) x A y B) → (𝐶 = (x𝐹y) → 𝐶 V))
873expb 1093 . . . 4 ((𝐹 Fn (A × B) (x A y B)) → (𝐶 = (x𝐹y) → 𝐶 V))
98rexlimdvva 2416 . . 3 (𝐹 Fn (A × B) → (x A y B 𝐶 = (x𝐹y) → 𝐶 V))
10 eqeq1 2028 . . . . . 6 (z = 𝐶 → (z = (x𝐹y) ↔ 𝐶 = (x𝐹y)))
11102rexbidv 2325 . . . . 5 (z = 𝐶 → (x A y B z = (x𝐹y) ↔ x A y B 𝐶 = (x𝐹y)))
1211elabg 2663 . . . 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 875   = wceq 1373   wcel 1375  {cab 2008  wrex 2283  Vcvv 2533   × cxp 4236  ran crn 4239   Fn wfn 4791  (class class class)co 5405
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-csb 2829  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-iun 3611  df-br 3717  df-opab 3771  df-mpt 3772  df-id 3983  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-iota 4761  df-fun 4798  df-fn 4799  df-fv 4804  df-ov 5408
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator