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Theorem oprssov 7602
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 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Proof of Theorem oprssov
StepHypRef Expression
1 ovres 7599 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
21adantl 481 . 2 (((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
3 fndm 6672 . . . . . . 7 (𝐺 Fn (𝐶 × 𝐷) → dom 𝐺 = (𝐶 × 𝐷))
43reseq2d 6000 . . . . . 6 (𝐺 Fn (𝐶 × 𝐷) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
543ad2ant2 1133 . . . . 5 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
6 funssres 6612 . . . . . 6 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
763adant2 1130 . . . . 5 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
85, 7eqtr3d 2777 . . . 4 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐹 ↾ (𝐶 × 𝐷)) = 𝐺)
98oveqd 7448 . . 3 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
109adantr 480 . 2 (((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
112, 10eqtr3d 2777 1 (((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wss 3963   × cxp 5687  dom cdm 5689  cres 5691  Fun wfun 6557   Fn wfn 6558  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434
This theorem is referenced by:  sspg  30757  ssps  30759
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