Theorem resdmres 5625

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 3824	. . . 4 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V)))
2		df-res 5126	. . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = (𝐴 ∩ (dom 𝐴 × V))
3		resdm2 5624	. . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴
4	2, 3	eqtr3i 2646	. . . . 5 ⊢ (𝐴 ∩ (dom 𝐴 × V)) = ◡◡𝐴
5	4	ineq2i 3811	. . . 4 ⊢ ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ ◡◡𝐴)
6		incom 3805	. . . 4 ⊢ ((𝐵 × V) ∩ ◡◡𝐴) = (◡◡𝐴 ∩ (𝐵 × V))
7	1, 5, 6	3eqtri 2648	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = (◡◡𝐴 ∩ (𝐵 × V))
8		df-res 5126	. . . 4 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V))
9		dmres 5419	. . . . . . 7 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
10	9	xpeq1i 5135	. . . . . 6 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 ∩ dom 𝐴) × V)
11		xpindir 5256	. . . . . 6 ⊢ ((𝐵 ∩ dom 𝐴) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
12	10, 11	eqtri 2644	. . . . 5 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
13	12	ineq2i 3811	. . . 4 ⊢ (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
14	8, 13	eqtri 2644	. . 3 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
15		df-res 5126	. . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (◡◡𝐴 ∩ (𝐵 × V))
16	7, 14, 15	3eqtr4i 2654	. 2 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (◡◡𝐴 ↾ 𝐵)
17		rescnvcnv 5597	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
18	16, 17	eqtri 2644	1 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1483  Vcvv 3200   ∩ cin 3573   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   ↾ cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  resresdm  5626  imadmres  5627  lindfres  20162  imacmp  21200  metreslem  22167  volres  23296  eccnvepres3  34050