Theorem imainrect 6173

Description: Image by a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (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 5681	. . 3 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
2	1	rneqi 5929	. 2 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3		df-ima 5682	. 2 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌)
4		df-ima 5682	. . . . 5 ⊢ (𝐺 “ (𝑌 ∩ 𝐴)) = ran (𝐺 ↾ (𝑌 ∩ 𝐴))
5		df-res 5681	. . . . . 6 ⊢ (𝐺 ↾ (𝑌 ∩ 𝐴)) = (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
6	5	rneqi 5929	. . . . 5 ⊢ ran (𝐺 ↾ (𝑌 ∩ 𝐴)) = ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
7	4, 6	eqtri 2754	. . . 4 ⊢ (𝐺 “ (𝑌 ∩ 𝐴)) = ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
8	7	ineq1i 4203	. . 3 ⊢ ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵) = (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
9		cnvin 6137	. . . . . 6 ⊢ ◡((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵))
10		inxp 5825	. . . . . . . . . 10 ⊢ ((𝐴 × V) ∩ (V × 𝐵)) = ((𝐴 ∩ V) × (V ∩ 𝐵))
11		inv1 4389	. . . . . . . . . . 11 ⊢ (𝐴 ∩ V) = 𝐴
12		incom 4196	. . . . . . . . . . . 12 ⊢ (V ∩ 𝐵) = (𝐵 ∩ V)
13		inv1 4389	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ V) = 𝐵
14	12, 13	eqtri 2754	. . . . . . . . . . 11 ⊢ (V ∩ 𝐵) = 𝐵
15	11, 14	xpeq12i 5697	. . . . . . . . . 10 ⊢ ((𝐴 ∩ V) × (V ∩ 𝐵)) = (𝐴 × 𝐵)
16	10, 15	eqtr2i 2755	. . . . . . . . 9 ⊢ (𝐴 × 𝐵) = ((𝐴 × V) ∩ (V × 𝐵))
17	16	ineq2i 4204	. . . . . . . 8 ⊢ ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
18		in32 4216	. . . . . . . 8 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵))
19		xpindir 5827	. . . . . . . . . . . 12 ⊢ ((𝑌 ∩ 𝐴) × V) = ((𝑌 × V) ∩ (𝐴 × V))
20	19	ineq2i 4204	. . . . . . . . . . 11 ⊢ (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
21		inass 4214	. . . . . . . . . . 11 ⊢ ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
22	20, 21	eqtr4i 2757	. . . . . . . . . 10 ⊢ (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V))
23	22	ineq1i 4203	. . . . . . . . 9 ⊢ ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵))
24		inass 4214	. . . . . . . . 9 ⊢ (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
25	23, 24	eqtri 2754	. . . . . . . 8 ⊢ ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
26	17, 18, 25	3eqtr4i 2764	. . . . . . 7 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵))
27	26	cnveqi 5867	. . . . . 6 ⊢ ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ◡((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵))
28		df-res 5681	. . . . . . 7 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (𝐵 × V))
29		cnvxp 6149	. . . . . . . 8 ⊢ ◡(V × 𝐵) = (𝐵 × V)
30	29	ineq2i 4204	. . . . . . 7 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵)) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (𝐵 × V))
31	28, 30	eqtr4i 2757	. . . . . 6 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵))
32	9, 27, 31	3eqtr4ri 2765	. . . . 5 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
33	32	dmeqi 5897	. . . 4 ⊢ dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = dom ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
34		incom 4196	. . . . 5 ⊢ (𝐵 ∩ dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V))) = (dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
35		dmres 5996	. . . . 5 ⊢ dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (𝐵 ∩ dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)))
36		df-rn 5680	. . . . . 6 ⊢ ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
37	36	ineq1i 4203	. . . . 5 ⊢ (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵) = (dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
38	34, 35, 37	3eqtr4ri 2765	. . . 4 ⊢ (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵) = dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵)
39		df-rn 5680	. . . 4 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = dom ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
40	33, 38, 39	3eqtr4ri 2765	. . 3 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
41	8, 40	eqtr4i 2757	. 2 ⊢ ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
42	2, 3, 41	3eqtr4i 2764	1 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵)

Syntax hints:   = wceq 1533  Vcvv 3468   ∩ cin 3942   × cxp 5667  ^◡ccnv 5668  dom cdm 5669  ran crn 5670   ↾ cres 5671   “ cima 5672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682

This theorem is referenced by:  ecinxp  8785  marypha1lem  9427