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Theorem oprssov 5584
Description: The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.)
Assertion
Ref Expression
oprssov (((Fun 𝐹 𝐺 Fn (𝐶 × 𝐷) 𝐺𝐹) (A 𝐶 B 𝐷)) → (A𝐹B) = (A𝐺B))

Proof of Theorem oprssov
StepHypRef Expression
1 ovres 5582 . . 3 ((A 𝐶 B 𝐷) → (A(𝐹 ↾ (𝐶 × 𝐷))B) = (A𝐹B))
21adantl 262 . 2 (((Fun 𝐹 𝐺 Fn (𝐶 × 𝐷) 𝐺𝐹) (A 𝐶 B 𝐷)) → (A(𝐹 ↾ (𝐶 × 𝐷))B) = (A𝐹B))
3 fndm 4941 . . . . . . 7 (𝐺 Fn (𝐶 × 𝐷) → dom 𝐺 = (𝐶 × 𝐷))
43reseq2d 4555 . . . . . 6 (𝐺 Fn (𝐶 × 𝐷) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
543ad2ant2 925 . . . . 5 ((Fun 𝐹 𝐺 Fn (𝐶 × 𝐷) 𝐺𝐹) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
6 funssres 4885 . . . . . 6 ((Fun 𝐹 𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
763adant2 922 . . . . 5 ((Fun 𝐹 𝐺 Fn (𝐶 × 𝐷) 𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
85, 7eqtr3d 2071 . . . 4 ((Fun 𝐹 𝐺 Fn (𝐶 × 𝐷) 𝐺𝐹) → (𝐹 ↾ (𝐶 × 𝐷)) = 𝐺)
98oveqd 5472 . . 3 ((Fun 𝐹 𝐺 Fn (𝐶 × 𝐷) 𝐺𝐹) → (A(𝐹 ↾ (𝐶 × 𝐷))B) = (A𝐺B))
109adantr 261 . 2 (((Fun 𝐹 𝐺 Fn (𝐶 × 𝐷) 𝐺𝐹) (A 𝐶 B 𝐷)) → (A(𝐹 ↾ (𝐶 × 𝐷))B) = (A𝐺B))
112, 10eqtr3d 2071 1 (((Fun 𝐹 𝐺 Fn (𝐶 × 𝐷) 𝐺𝐹) (A 𝐶 B 𝐷)) → (A𝐹B) = (A𝐺B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wss 2911   × cxp 4286  dom cdm 4288  cres 4290  Fun wfun 4839   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-bnd 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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  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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