Theorem oprssov 7561

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 7558	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
2	1	adantl 485	. 2 ⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵))
3		fndm 6620	. . . . . . 7 ⊢ (𝐺 Fn (𝐶 × 𝐷) → dom 𝐺 = (𝐶 × 𝐷))
4	3	reseq2d 5963	. . . . . 6 ⊢ (𝐺 Fn (𝐶 × 𝐷) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
5	4	3ad2ant2 1146	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = (𝐹 ↾ (𝐶 × 𝐷)))
6		funssres 6561	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
7	6	3adant2 1143	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
8	5, 7	eqtr3d 2798	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ (𝐶 × 𝐷)) = 𝐺)
9	8	oveqd 7409	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
10	9	adantr 484	. 2 ⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐺𝐵))
11	2, 10	eqtr3d 2798	1 ⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ⊆ wss 3904   × cxp 5643  dom cdm 5645   ↾ cres 5647  Fun wfun 6511   Fn wfn 6512  (class class class)co 7392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-res 5657  df-iota 6473  df-fun 6519  df-fn 6520  df-fv 6525  df-ov 7395

This theorem is referenced by:  sspg  30877  ssps  30879