Theorem ranncnt 14625

Description: Range of the intersection of the inclusion with a square cross product.

Hypothesis

Ref	Expression
ranncnt.1	|- C = {<.x, y>. | x C_ y}

Assertion

Ref	Expression
ranncnt	|- ran ( C i^i (A X. A)) = A

Distinct variable group:   x,A,y

Proof of Theorem ranncnt

Step	Hyp	Ref	Expression
1		ranncnt.1	. . . 4 |- C = {<.x, y>. | x C_ y}
2		df-xp 4000	. . . 4 |- (A X. A) = {<.x, y>. | (x e. A /\ y e. A)}
3	1, 2	ineq12i 2794	. . 3 |- (C i^i (A X. A)) = ({<.x, y>. | x C_ y} i^i {<.x, y>. | (x e. A /\ y e. A)})
4	3	rneqi 4187	. 2 |- ran ( C i^i (A X. A)) = ran ({<.x, y>. | x C_ y} i^i {<.x, y>. | (x e. A /\ y e. A)})
5		incom 2787	. . . 4 |- ({<.x, y>. | x C_ y} i^i {<.x, y>. | (x e. A /\ y e. A)}) = ({<.x, y>. | (x e. A /\ y e. A)} i^i {<.x, y>. | x C_ y})
6		inopab 4108	. . . 4 |- ({<.x, y>. | (x e. A /\ y e. A)} i^i {<.x, y>. | x C_ y}) = {<.x, y>. | ((x e. A /\ y e. A) /\ x C_ y)}
7	5, 6	eqtri 1908	. . 3 |- ({<.x, y>. | x C_ y} i^i {<.x, y>. | (x e. A /\ y e. A)}) = {<.x, y>. | ((x e. A /\ y e. A) /\ x C_ y)}
8	7	rneqi 4187	. 2 |- ran ({<.x, y>. | x C_ y} i^i {<.x, y>. | (x e. A /\ y e. A)}) = ran {<.x, y>. | ((x e. A /\ y e. A) /\ x C_ y)}
9		df-rn 4005	. . 3 |- ran {<.x, y>. | ((x e. A /\ y e. A) /\ x C_ y)} = dom `'{<.x, y>. | ((x e. A /\ y e. A) /\ x C_ y)}
10		cnvopab 4317	. . . 4 |- `'{<.x, y>. | ((x e. A /\ y e. A) /\ x C_ y)} = {<.y, x>. | ((x e. A /\ y e. A) /\ x C_ y)}
11	10	dmeqi 4158	. . 3 |- dom `'{<.x, y>. | ((x e. A /\ y e. A) /\ x C_ y)} = dom {<.y, x>. | ((x e. A /\ y e. A) /\ x C_ y)}
12		simplr 449	. . . . . . . 8 |- (((x e. A /\ y e. A) /\ x C_ y) -> y e. A)
13		simpll 448	. . . . . . . . 9 |- (((x e. A /\ y e. A) /\ x C_ y) -> x e. A)
14		simpr 350	. . . . . . . . 9 |- (((x e. A /\ y e. A) /\ x C_ y) -> x C_ y)
15	13, 14	jca 310	. . . . . . . 8 |- (((x e. A /\ y e. A) /\ x C_ y) -> (x e. A /\ x C_ y))
16	12, 15	jca 310	. . . . . . 7 |- (((x e. A /\ y e. A) /\ x C_ y) -> (y e. A /\ (x e. A /\ x C_ y)))
17		simprl 450	. . . . . . . . 9 |- ((y e. A /\ (x e. A /\ x C_ y)) -> x e. A)
18		simpl 346	. . . . . . . . 9 |- ((y e. A /\ (x e. A /\ x C_ y)) -> y e. A)
19	17, 18	jca 310	. . . . . . . 8 |- ((y e. A /\ (x e. A /\ x C_ y)) -> (x e. A /\ y e. A))
20		simprr 451	. . . . . . . 8 |- ((y e. A /\ (x e. A /\ x C_ y)) -> x C_ y)
21	19, 20	jca 310	. . . . . . 7 |- ((y e. A /\ (x e. A /\ x C_ y)) -> ((x e. A /\ y e. A) /\ x C_ y))
22	16, 21	impbii 174	. . . . . 6 |- (((x e. A /\ y e. A) /\ x C_ y) <-> (y e. A /\ (x e. A /\ x C_ y)))
23	22	opabbii 3402	. . . . 5 |- {<.y, x>. | ((x e. A /\ y e. A) /\ x C_ y)} = {<.y, x>. | (y e. A /\ (x e. A /\ x C_ y))}
24	23	dmeqi 4158	. . . 4 |- dom {<.y, x>. | ((x e. A /\ y e. A) /\ x C_ y)} = dom {<.y, x>. | (y e. A /\ (x e. A /\ x C_ y))}
25		ssid 2634	. . . . . . 7 |- y C_ y
26		sseq1 2637	. . . . . . . . . 10 |- (x = y -> (x C_ y <-> y C_ y))
27	26	rcla4ev 2381	. . . . . . . . 9 |- ((y e. A /\ y C_ y) -> E.x e. A x C_ y)
28	27	ancoms 484	. . . . . . . 8 |- ((y C_ y /\ y e. A) -> E.x e. A x C_ y)
29		df-rex 2110	. . . . . . . 8 |- (E.x e. A x C_ y <-> E.x(x e. A /\ x C_ y))
30	28, 29	sylib 215	. . . . . . 7 |- ((y C_ y /\ y e. A) -> E.x(x e. A /\ x C_ y))
31	25, 30	mpan 759	. . . . . 6 |- (y e. A -> E.x(x e. A /\ x C_ y))
32	31	rgen 2159	. . . . 5 |- A.y e. A E.x(x e. A /\ x C_ y)
33		dmopab3 4169	. . . . 5 |- (A.y e. A E.x(x e. A /\ x C_ y) <-> dom {<.y, x>. | (y e. A /\ (x e. A /\ x C_ y))} = A)
34	32, 33	mpbi 206	. . . 4 |- dom {<.y, x>. | (y e. A /\ (x e. A /\ x C_ y))} = A
35	24, 34	eqtri 1908	. . 3 |- dom {<.y, x>. | ((x e. A /\ y e. A) /\ x C_ y)} = A
36	9, 11, 35	3eqtri 1912	. 2 |- ran {<.x, y>. | ((x e. A /\ y e. A) /\ x C_ y)} = A
37	4, 8, 36	3eqtri 1912	1 |- ran ( C i^i (A X. A)) = A

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593  {copab 3395   X. cxp 3984  `'ccnv 3985  dom cdm 3986  ran crn 3987

This theorem is referenced by:  toplat 14638

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005

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