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Theorem oprssov 7311
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 7308 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
21adantl 482 . 2 (((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
3 fndm 6454 . . . . . . 7 (𝐺 Fn (𝐶 × 𝐷) → dom 𝐺 = (𝐶 × 𝐷))
43reseq2d 5852 . . . . . 6 (𝐺 Fn (𝐶 × 𝐷) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
543ad2ant2 1128 . . . . 5 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
6 funssres 6397 . . . . . 6 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
763adant2 1125 . . . . 5 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
85, 7eqtr3d 2863 . . . 4 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐹 ↾ (𝐶 × 𝐷)) = 𝐺)
98oveqd 7167 . . 3 ((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
109adantr 481 . 2 (((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
112, 10eqtr3d 2863 1 (((Fun 𝐹𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺𝐹) ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  wss 3940   × cxp 5552  dom cdm 5554  cres 5556  Fun wfun 6348   Fn wfn 6349  (class class class)co 7150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-res 5566  df-iota 6313  df-fun 6356  df-fn 6357  df-fv 6362  df-ov 7153
This theorem is referenced by:  sspg  28438  ssps  28440
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