Theorem psssdm2 17215

Description: Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)

Hypothesis

Ref	Expression
psssdm.1	|- X = dom R

Assertion

Ref	Expression
psssdm2	|- ( R e. PosetRel -> dom ( R i^i ( A X. A ) ) = ( X i^i A ) )

Proof of Theorem psssdm2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmin 5332	. . . 4 |- dom ( R i^i ( A X. A ) ) C_ ( dom R i^i dom ( A X. A ) )
2		psssdm.1	. . . . . 6 |- X = dom R
3	2	eqcomi 2631	. . . . 5 |- dom R = X
4		dmxpid 5345	. . . . 5 |- dom ( A X. A ) = A
5	3, 4	ineq12i 3812	. . . 4 |- ( dom R i^i dom ( A X. A ) ) = ( X i^i A )
6	1, 5	sseqtri 3637	. . 3 |- dom ( R i^i ( A X. A ) ) C_ ( X i^i A )
7	6	a1i 11	. 2 |- ( R e. PosetRel -> dom ( R i^i ( A X. A ) ) C_ ( X i^i A ) )
8		inss2 3834	. . . . . . 7 |- ( X i^i A ) C_ A
9		simpr 477	. . . . . . 7 |- ( ( R e. PosetRel /\ x e. ( X i^i A ) ) -> x e. ( X i^i A ) )
10	8, 9	sseldi 3601	. . . . . 6 |- ( ( R e. PosetRel /\ x e. ( X i^i A ) ) -> x e. A )
11		inss1 3833	. . . . . . . 8 |- ( X i^i A ) C_ X
12	11	sseli 3599	. . . . . . 7 |- ( x e. ( X i^i A ) -> x e. X )
13	2	psref 17208	. . . . . . 7 |- ( ( R e. PosetRel /\ x e. X ) -> x R x )
14	12, 13	sylan2 491	. . . . . 6 |- ( ( R e. PosetRel /\ x e. ( X i^i A ) ) -> x R x )
15		brinxp2 5180	. . . . . 6 |- ( x ( R i^i ( A X. A ) ) x <-> ( x e. A /\ x e. A /\ x R x ) )
16	10, 10, 14, 15	syl3anbrc 1246	. . . . 5 |- ( ( R e. PosetRel /\ x e. ( X i^i A ) ) -> x ( R i^i ( A X. A ) ) x )
17		vex 3203	. . . . . 6 |- x e. _V
18	17, 17	breldm 5329	. . . . 5 |- ( x ( R i^i ( A X. A ) ) x -> x e. dom ( R i^i ( A X. A ) ) )
19	16, 18	syl 17	. . . 4 |- ( ( R e. PosetRel /\ x e. ( X i^i A ) ) -> x e. dom ( R i^i ( A X. A ) ) )
20	19	ex 450	. . 3 |- ( R e. PosetRel -> ( x e. ( X i^i A ) -> x e. dom ( R i^i ( A X. A ) ) ) )
21	20	ssrdv 3609	. 2 |- ( R e. PosetRel -> ( X i^i A ) C_ dom ( R i^i ( A X. A ) ) )
22	7, 21	eqssd 3620	1 |- ( R e. PosetRel -> dom ( R i^i ( A X. A ) ) = ( X i^i A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   class class class wbr 4653   X. cxp 5112   dom cdm 5114   PosetRelcps 17198

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-ne 2795  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-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ps 17200

This theorem is referenced by:  psssdm  17216  ordtrest  21006