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Theorem oprssov 6970
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 6967 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
21adantl 473 . 2 (((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
3 fndm 6152 . . . . . . 7 (𝐺 Fn (𝐶 × 𝐷) → dom 𝐺 = (𝐶 × 𝐷))
43reseq2d 5552 . . . . . 6 (𝐺 Fn (𝐶 × 𝐷) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
543ad2ant2 1129 . . . . 5 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
6 funssres 6092 . . . . . 6 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
763adant2 1126 . . . . 5 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
85, 7eqtr3d 2797 . . . 4 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐹 ↾ (𝐶 × 𝐷)) = 𝐺)
98oveqd 6832 . . 3 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
109adantr 472 . 2 (((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
112, 10eqtr3d 2797 1 (((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2140  wss 3716   × cxp 5265  dom cdm 5267  cres 5269  Fun wfun 6044   Fn wfn 6045  (class class class)co 6815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-res 5279  df-iota 6013  df-fun 6052  df-fn 6053  df-fv 6058  df-ov 6818
This theorem is referenced by:  sspg  27914  ssps  27916
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