Theorem oprssov 7569

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 7566	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
2	1	adantl 486	. 2 ⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
3		fndm 6628	. . . . . . 7 ⊢ (𝐺 Fn (𝐶 × 𝐷) → dom 𝐺 = (𝐶 × 𝐷))
4	3	reseq2d 5969	. . . . . 6 ⊢ (𝐺 Fn (𝐶 × 𝐷) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
5	4	3ad2ant2 1150	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
6		funssres 6569	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
7	6	3adant2 1147	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
8	5, 7	eqtr3d 2802	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ (𝐶 × 𝐷)) = 𝐺)
9	8	oveqd 7417	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
10	9	adantr 485	. 2 ⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
11	2, 10	eqtr3d 2802	1 ⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ⊆ wss 3907   × cxp 5650  dom cdm 5652   ↾ cres 5654  Fun wfun 6519   Fn wfn 6520  (class class class)co 7400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-res 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-fv 6533  df-ov 7403

This theorem is referenced by:  sspg  30989  ssps  30991