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Theorem ovelrn 5591
 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 5588 . . 3 (𝐹 Fn (A × B) → ran 𝐹 = {zx A y B z = (x𝐹y)})
21eleq2d 2104 . 2 (𝐹 Fn (A × B) → (𝐶 ran 𝐹𝐶 {zx A y B z = (x𝐹y)}))
3 elex 2560 . . . 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 5481 . . . . . 6 ((𝐹 Fn (A × B) x A y B) → (x𝐹y) V)
6 eleq1 2097 . . . . . 6 (𝐶 = (x𝐹y) → (𝐶 V ↔ (x𝐹y) V))
75, 6syl5ibrcom 146 . . . . 5 ((𝐹 Fn (A × B) x A y B) → (𝐶 = (x𝐹y) → 𝐶 V))
873expb 1104 . . . 4 ((𝐹 Fn (A × B) (x A y B)) → (𝐶 = (x𝐹y) → 𝐶 V))
98rexlimdvva 2434 . . 3 (𝐹 Fn (A × B) → (x A y B 𝐶 = (x𝐹y) → 𝐶 V))
10 eqeq1 2043 . . . . . 6 (z = 𝐶 → (z = (x𝐹y) ↔ 𝐶 = (x𝐹y)))
11102rexbidv 2343 . . . . 5 (z = 𝐶 → (x A y B z = (x𝐹y) ↔ x A y B 𝐶 = (x𝐹y)))
1211elabg 2682 . . . 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 620 . 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 884   = wceq 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551   × cxp 4286  ran crn 4289   Fn wfn 4840  (class class class)co 5455 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-ov 5458 This theorem is referenced by: (None)
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