Theorem imainrect 5494

Description: Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Assertion

Ref	Expression
imainrect	⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵)

Proof of Theorem imainrect

Step	Hyp	Ref	Expression
1		df-res 5050	. . 3 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
2	1	rneqi 5273	. 2 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3		df-ima 5051	. 2 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌)
4		df-ima 5051	. . . . 5 ⊢ (𝐺 “ (𝑌 ∩ 𝐴)) = ran (𝐺 ↾ (𝑌 ∩ 𝐴))
5		df-res 5050	. . . . . 6 ⊢ (𝐺 ↾ (𝑌 ∩ 𝐴)) = (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
6	5	rneqi 5273	. . . . 5 ⊢ ran (𝐺 ↾ (𝑌 ∩ 𝐴)) = ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
7	4, 6	eqtri 2632	. . . 4 ⊢ (𝐺 “ (𝑌 ∩ 𝐴)) = ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
8	7	ineq1i 3772	. . 3 ⊢ ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵) = (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
9		cnvin 5459	. . . . . 6 ⊢ ◡((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵))
10		inxp 5176	. . . . . . . . . 10 ⊢ ((𝐴 × V) ∩ (V × 𝐵)) = ((𝐴 ∩ V) × (V ∩ 𝐵))
11		inv1 3922	. . . . . . . . . . 11 ⊢ (𝐴 ∩ V) = 𝐴
12		incom 3767	. . . . . . . . . . . 12 ⊢ (V ∩ 𝐵) = (𝐵 ∩ V)
13		inv1 3922	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ V) = 𝐵
14	12, 13	eqtri 2632	. . . . . . . . . . 11 ⊢ (V ∩ 𝐵) = 𝐵
15	11, 14	xpeq12i 5061	. . . . . . . . . 10 ⊢ ((𝐴 ∩ V) × (V ∩ 𝐵)) = (𝐴 × 𝐵)
16	10, 15	eqtr2i 2633	. . . . . . . . 9 ⊢ (𝐴 × 𝐵) = ((𝐴 × V) ∩ (V × 𝐵))
17	16	ineq2i 3773	. . . . . . . 8 ⊢ ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
18		in32 3787	. . . . . . . 8 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵))
19		xpindir 5178	. . . . . . . . . . . 12 ⊢ ((𝑌 ∩ 𝐴) × V) = ((𝑌 × V) ∩ (𝐴 × V))
20	19	ineq2i 3773	. . . . . . . . . . 11 ⊢ (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
21		inass 3785	. . . . . . . . . . 11 ⊢ ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
22	20, 21	eqtr4i 2635	. . . . . . . . . 10 ⊢ (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V))
23	22	ineq1i 3772	. . . . . . . . 9 ⊢ ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵))
24		inass 3785	. . . . . . . . 9 ⊢ (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
25	23, 24	eqtri 2632	. . . . . . . 8 ⊢ ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
26	17, 18, 25	3eqtr4i 2642	. . . . . . 7 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵))
27	26	cnveqi 5219	. . . . . 6 ⊢ ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ◡((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵))
28		df-res 5050	. . . . . . 7 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (𝐵 × V))
29		cnvxp 5470	. . . . . . . 8 ⊢ ◡(V × 𝐵) = (𝐵 × V)
30	29	ineq2i 3773	. . . . . . 7 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵)) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (𝐵 × V))
31	28, 30	eqtr4i 2635	. . . . . 6 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵))
32	9, 27, 31	3eqtr4ri 2643	. . . . 5 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
33	32	dmeqi 5247	. . . 4 ⊢ dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = dom ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
34		incom 3767	. . . . 5 ⊢ (𝐵 ∩ dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V))) = (dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
35		dmres 5339	. . . . 5 ⊢ dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (𝐵 ∩ dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)))
36		df-rn 5049	. . . . . 6 ⊢ ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
37	36	ineq1i 3772	. . . . 5 ⊢ (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵) = (dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
38	34, 35, 37	3eqtr4ri 2643	. . . 4 ⊢ (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵) = dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵)
39		df-rn 5049	. . . 4 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = dom ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
40	33, 38, 39	3eqtr4ri 2643	. . 3 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
41	8, 40	eqtr4i 2635	. 2 ⊢ ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
42	2, 3, 41	3eqtr4i 2642	1 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵)

Syntax hints:   = wceq 1475  Vcvv 3173   ∩ cin 3539   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051

This theorem is referenced by:  ecinxp  7709  marypha1lem  8222