Theorem ustuni 22030

Description: The set union of a uniform structure is the Cartesian product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.)

Assertion

Ref	Expression
ustuni	|- ( U e. (UnifOn ` X ) -> U. U = ( X X. X ) )

Proof of Theorem ustuni

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ustbasel 22010	. 2 |- ( U e. (UnifOn ` X ) -> ( X X. X ) e. U )
2		ustssxp 22008	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ u e. U ) -> u C_ ( X X. X ) )
3	2	ralrimiva 2966	. . 3 |- ( U e. (UnifOn ` X ) -> A. u e. U u C_ ( X X. X ) )
4		pwssb 4612	. . 3 |- ( U C_ ~P ( X X. X ) <-> A. u e. U u C_ ( X X. X ) )
5	3, 4	sylibr 224	. 2 |- ( U e. (UnifOn ` X ) -> U C_ ~P ( X X. X ) )
6		elpwuni 4616	. . 3 |- ( ( X X. X ) e. U -> ( U C_ ~P ( X X. X ) <-> U. U = ( X X. X ) ) )
7	6	biimpa 501	. 2 |- ( ( ( X X. X ) e. U /\ U C_ ~P ( X X. X ) ) -> U. U = ( X X. X ) )
8	1, 5, 7	syl2anc 693	1 |- ( U e. (UnifOn ` X ) -> U. U = ( X X. X ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   X. cxp 5112   ` cfv 5888  UnifOncust 22003

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ust 22004

This theorem is referenced by:  tususs  22074  cnflduss  23152