Theorem resdmres 5188

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 3388	. . . 4 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V)))
2		df-res 4700	. . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = (𝐴 ∩ (dom 𝐴 × V))
3		resdm2 5187	. . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴
4	2, 3	eqtr3i 2229	. . . . 5 ⊢ (𝐴 ∩ (dom 𝐴 × V)) = ◡◡𝐴
5	4	ineq2i 3375	. . . 4 ⊢ ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ ◡◡𝐴)
6		incom 3369	. . . 4 ⊢ ((𝐵 × V) ∩ ◡◡𝐴) = (◡◡𝐴 ∩ (𝐵 × V))
7	1, 5, 6	3eqtri 2231	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = (◡◡𝐴 ∩ (𝐵 × V))
8		df-res 4700	. . . 4 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V))
9		dmres 4994	. . . . . . 7 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
10	9	xpeq1i 4708	. . . . . 6 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 ∩ dom 𝐴) × V)
11		xpindir 4827	. . . . . 6 ⊢ ((𝐵 ∩ dom 𝐴) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
12	10, 11	eqtri 2227	. . . . 5 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
13	12	ineq2i 3375	. . . 4 ⊢ (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
14	8, 13	eqtri 2227	. . 3 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
15		df-res 4700	. . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (◡◡𝐴 ∩ (𝐵 × V))
16	7, 14, 15	3eqtr4i 2237	. 2 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (◡◡𝐴 ↾ 𝐵)
17		rescnvcnv 5159	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
18	16, 17	eqtri 2227	1 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1373  Vcvv 2773   ∩ cin 3169   × cxp 4686  ^◡ccnv 4687  dom cdm 4688   ↾ cres 4690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4055  df-opab 4117  df-xp 4694  df-rel 4695  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700

This theorem is referenced by:  imadmres  5189  metreslem  14937