Theorem resdmres 6254

Description: Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
resdmres	⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)

Proof of Theorem resdmres

Step	Hyp	Ref	Expression
1		in12 4237	. . . 4 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V)))
2		df-res 5701	. . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = (𝐴 ∩ (dom 𝐴 × V))
3		resdm2 6253	. . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴
4	2, 3	eqtr3i 2765	. . . . 5 ⊢ (𝐴 ∩ (dom 𝐴 × V)) = ◡◡𝐴
5	4	ineq2i 4225	. . . 4 ⊢ ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ ◡◡𝐴)
6		incom 4217	. . . 4 ⊢ ((𝐵 × V) ∩ ◡◡𝐴) = (◡◡𝐴 ∩ (𝐵 × V))
7	1, 5, 6	3eqtri 2767	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = (◡◡𝐴 ∩ (𝐵 × V))
8		df-res 5701	. . . 4 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V))
9		dmres 6032	. . . . . . 7 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
10	9	xpeq1i 5715	. . . . . 6 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 ∩ dom 𝐴) × V)
11		xpindir 5848	. . . . . 6 ⊢ ((𝐵 ∩ dom 𝐴) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
12	10, 11	eqtri 2763	. . . . 5 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
13	12	ineq2i 4225	. . . 4 ⊢ (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
14	8, 13	eqtri 2763	. . 3 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
15		df-res 5701	. . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (◡◡𝐴 ∩ (𝐵 × V))
16	7, 14, 15	3eqtr4i 2773	. 2 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (◡◡𝐴 ↾ 𝐵)
17		rescnvcnv 6226	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
18	16, 17	eqtri 2763	1 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1537  Vcvv 3478   ∩ cin 3962   × cxp 5687  ^◡ccnv 5688  dom cdm 5689   ↾ cres 5691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701

This theorem is referenced by:  resresdm  6255  imadmres  6256  lindfres  21861  imacmp  23421  metreslem  24388  volres  25577  eccnvepres3  38268  isubgruhgr  47792