Theorem oprssov 6701

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) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Proof of Theorem oprssov

Step	Hyp	Ref	Expression
1		ovres 6698	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
2	1	adantl 481	. 2 ⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
3		fndm 5904	. . . . . . 7 ⊢ (𝐺 Fn (𝐶 × 𝐷) → dom 𝐺 = (𝐶 × 𝐷))
4	3	reseq2d 5317	. . . . . 6 ⊢ (𝐺 Fn (𝐶 × 𝐷) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
5	4	3ad2ant2 1076	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
6		funssres 5844	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
7	6	3adant2 1073	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
8	5, 7	eqtr3d 2646	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ (𝐶 × 𝐷)) = 𝐺)
9	8	oveqd 6566	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
10	9	adantr 480	. 2 ⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
11	2, 10	eqtr3d 2646	1 ⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   × cxp 5036  dom cdm 5038   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552

This theorem is referenced by:  sspg  26967  ssps  26969