Theorem unxpwdom 8494

Description: If a Cartesian product is dominated by a union, then the base set is either weakly dominated by one factor of the union or dominated by the other. Extracted from Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
unxpwdom	|- ( ( A X. A ) ~<_ ( B u. C ) -> ( A ~<_* B \/ A ~<_ C ) )

Proof of Theorem unxpwdom

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 7961	. . . . 5 |- Rel ~<_
2	1	brrelex2i 5159	. . . 4 |- ( ( A X. A ) ~<_ ( B u. C ) -> ( B u. C ) e. _V )
3		domeng 7969	. . . 4 |- ( ( B u. C ) e. _V -> ( ( A X. A ) ~<_ ( B u. C ) <-> E. x ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) )
4	2, 3	syl 17	. . 3 |- ( ( A X. A ) ~<_ ( B u. C ) -> ( ( A X. A ) ~<_ ( B u. C ) <-> E. x ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) )
5	4	ibi 256	. 2 |- ( ( A X. A ) ~<_ ( B u. C ) -> E. x ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) )
6		simprl 794	. . . . 5 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( A X. A ) ~~ x )
7		indi 3873	. . . . . 6 |- ( x i^i ( B u. C ) ) = ( ( x i^i B ) u. ( x i^i C ) )
8		simprr 796	. . . . . . 7 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> x C_ ( B u. C ) )
9		df-ss 3588	. . . . . . 7 |- ( x C_ ( B u. C ) <-> ( x i^i ( B u. C ) ) = x )
10	8, 9	sylib 208	. . . . . 6 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( x i^i ( B u. C ) ) = x )
11	7, 10	syl5eqr 2670	. . . . 5 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( ( x i^i B ) u. ( x i^i C ) ) = x )
12	6, 11	breqtrrd 4681	. . . 4 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( A X. A ) ~~ ( ( x i^i B ) u. ( x i^i C ) ) )
13		unxpwdom2 8493	. . . 4 |- ( ( A X. A ) ~~ ( ( x i^i B ) u. ( x i^i C ) ) -> ( A ~<_* ( x i^i B ) \/ A ~<_ ( x i^i C ) ) )
14	12, 13	syl 17	. . 3 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( A ~<_* ( x i^i B ) \/ A ~<_ ( x i^i C ) ) )
15		ssun1 3776	. . . . . . . 8 |- B C_ ( B u. C )
16	2	adantr 481	. . . . . . . 8 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( B u. C ) e. _V )
17		ssexg 4804	. . . . . . . 8 |- ( ( B C_ ( B u. C ) /\ ( B u. C ) e. _V ) -> B e. _V )
18	15, 16, 17	sylancr 695	. . . . . . 7 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> B e. _V )
19		inss2 3834	. . . . . . 7 |- ( x i^i B ) C_ B
20		ssdomg 8001	. . . . . . 7 |- ( B e. _V -> ( ( x i^i B ) C_ B -> ( x i^i B ) ~<_ B ) )
21	18, 19, 20	mpisyl 21	. . . . . 6 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( x i^i B ) ~<_ B )
22		domwdom 8479	. . . . . 6 |- ( ( x i^i B ) ~<_ B -> ( x i^i B ) ~<_* B )
23	21, 22	syl 17	. . . . 5 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( x i^i B ) ~<_* B )
24		wdomtr 8480	. . . . . 6 |- ( ( A ~<_* ( x i^i B ) /\ ( x i^i B ) ~<_* B ) -> A ~<_* B )
25	24	expcom 451	. . . . 5 |- ( ( x i^i B ) ~<_* B -> ( A ~<_* ( x i^i B ) -> A ~<_* B ) )
26	23, 25	syl 17	. . . 4 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( A ~<_* ( x i^i B ) -> A ~<_* B ) )
27		ssun2 3777	. . . . . . 7 |- C C_ ( B u. C )
28		ssexg 4804	. . . . . . 7 |- ( ( C C_ ( B u. C ) /\ ( B u. C ) e. _V ) -> C e. _V )
29	27, 16, 28	sylancr 695	. . . . . 6 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> C e. _V )
30		inss2 3834	. . . . . 6 |- ( x i^i C ) C_ C
31		ssdomg 8001	. . . . . 6 |- ( C e. _V -> ( ( x i^i C ) C_ C -> ( x i^i C ) ~<_ C ) )
32	29, 30, 31	mpisyl 21	. . . . 5 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( x i^i C ) ~<_ C )
33		domtr 8009	. . . . . 6 |- ( ( A ~<_ ( x i^i C ) /\ ( x i^i C ) ~<_ C ) -> A ~<_ C )
34	33	expcom 451	. . . . 5 |- ( ( x i^i C ) ~<_ C -> ( A ~<_ ( x i^i C ) -> A ~<_ C ) )
35	32, 34	syl 17	. . . 4 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( A ~<_ ( x i^i C ) -> A ~<_ C ) )
36	26, 35	orim12d 883	. . 3 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( ( A ~<_* ( x i^i B ) \/ A ~<_ ( x i^i C ) ) -> ( A ~<_* B \/ A ~<_ C ) ) )
37	14, 36	mpd 15	. 2 |- ( ( ( A X. A ) ~<_ ( B u. C ) /\ ( ( A X. A ) ~~ x /\ x C_ ( B u. C ) ) ) -> ( A ~<_* B \/ A ~<_ C ) )
38	5, 37	exlimddv 1863	1 |- ( ( A X. A ) ~<_ ( B u. C ) -> ( A ~<_* B \/ A ~<_ C ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   class class class wbr 4653   X. cxp 5112   ~~ cen 7952   ~<_ cdom 7953   ~<_^* cwdom 8462

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-int 4476  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-wdom 8464

This theorem is referenced by:  pwcdadom  9038