Theorem djudom 7291

Description: Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)

Assertion

Ref	Expression
djudom	|- ( ( A ~<_ B /\ C ~<_ D ) -> ( A ⊔ C ) ~<_ ( B ⊔ D ) )

Proof of Theorem djudom

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 6919	. . 3 |- ( A ~<_ B -> E. f f : A -1-1-> B )
2	1	adantr 276	. 2 |- ( ( A ~<_ B /\ C ~<_ D ) -> E. f f : A -1-1-> B )
3		brdomi 6919	. . . 4 |- ( C ~<_ D -> E. g g : C -1-1-> D )
4	3	ad2antlr 489	. . 3 |- ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) -> E. g g : C -1-1-> D )
5		inlresf1 7259	. . . . . . . . 9 |- (inl |` B ) : B -1-1-> ( B ⊔ D )
6		simplr 529	. . . . . . . . 9 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> f : A -1-1-> B )
7		f1co 5554	. . . . . . . . 9 |- ( ( (inl |` B ) : B -1-1-> ( B ⊔ D ) /\ f : A -1-1-> B ) -> ( (inl |` B ) o. f ) : A -1-1-> ( B ⊔ D ) )
8	5, 6, 7	sylancr 414	. . . . . . . 8 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ( (inl |` B ) o. f ) : A -1-1-> ( B ⊔ D ) )
9		inrresf1 7260	. . . . . . . . 9 |- (inr |` D ) : D -1-1-> ( B ⊔ D )
10		simpr 110	. . . . . . . . 9 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> g : C -1-1-> D )
11		f1co 5554	. . . . . . . . 9 |- ( ( (inr |` D ) : D -1-1-> ( B ⊔ D ) /\ g : C -1-1-> D ) -> ( (inr |` D ) o. g ) : C -1-1-> ( B ⊔ D ) )
12	9, 10, 11	sylancr 414	. . . . . . . 8 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ( (inr |` D ) o. g ) : C -1-1-> ( B ⊔ D ) )
13		rnco 5243	. . . . . . . . . . 11 |- ran ( (inl |` B ) o. f ) = ran ( (inl |` B ) |` ran f )
14		f1rn 5543	. . . . . . . . . . . . . 14 |- ( f : A -1-1-> B -> ran f C_ B )
15	14	ad2antlr 489	. . . . . . . . . . . . 13 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ran f C_ B )
16		resabs1 5042	. . . . . . . . . . . . 13 |- ( ran f C_ B -> ( (inl |` B ) |` ran f ) = (inl |` ran f ) )
17	15, 16	syl 14	. . . . . . . . . . . 12 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ( (inl |` B ) |` ran f ) = (inl |` ran f ) )
18	17	rneqd 4961	. . . . . . . . . . 11 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ran ( (inl |` B ) |` ran f ) = ran (inl |` ran f ) )
19	13, 18	eqtrid 2276	. . . . . . . . . 10 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ran ( (inl |` B ) o. f ) = ran (inl |` ran f ) )
20		rnco 5243	. . . . . . . . . . 11 |- ran ( (inr |` D ) o. g ) = ran ( (inr |` D ) |` ran g )
21		f1rn 5543	. . . . . . . . . . . . 13 |- ( g : C -1-1-> D -> ran g C_ D )
22		resabs1 5042	. . . . . . . . . . . . 13 |- ( ran g C_ D -> ( (inr |` D ) |` ran g ) = (inr |` ran g ) )
23	10, 21, 22	3syl 17	. . . . . . . . . . . 12 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ( (inr |` D ) |` ran g ) = (inr |` ran g ) )
24	23	rneqd 4961	. . . . . . . . . . 11 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ran ( (inr |` D ) |` ran g ) = ran (inr |` ran g ) )
25	20, 24	eqtrid 2276	. . . . . . . . . 10 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ran ( (inr |` D ) o. g ) = ran (inr |` ran g ) )
26	19, 25	ineq12d 3409	. . . . . . . . 9 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ( ran ( (inl |` B ) o. f ) i^i ran ( (inr |` D ) o. g ) ) = ( ran (inl |` ran f ) i^i ran (inr |` ran g ) ) )
27		djuinr 7261	. . . . . . . . 9 |- ( ran (inl |` ran f ) i^i ran (inr |` ran g ) ) = (/)
28	26, 27	eqtrdi 2280	. . . . . . . 8 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ( ran ( (inl |` B ) o. f ) i^i ran ( (inr |` D ) o. g ) ) = (/) )
29	8, 12, 28	casef1 7288	. . . . . . 7 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> case ( ( (inl |` B ) o. f ) , ( (inr |` D ) o. g ) ) : ( A ⊔ C ) -1-1-> ( B ⊔ D ) )
30		f1f 5542	. . . . . . 7 |- (case ( ( (inl |` B ) o. f ) , ( (inr |` D ) o. g ) ) : ( A ⊔ C ) -1-1-> ( B ⊔ D ) -> case ( ( (inl |` B ) o. f ) , ( (inr |` D ) o. g ) ) : ( A ⊔ C ) --> ( B ⊔ D ) )
31	29, 30	syl 14	. . . . . 6 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> case ( ( (inl |` B ) o. f ) , ( (inr |` D ) o. g ) ) : ( A ⊔ C ) --> ( B ⊔ D ) )
32		reldom 6913	. . . . . . . . 9 |- Rel ~<_
33	32	brrelex1i 4769	. . . . . . . 8 |- ( A ~<_ B -> A e. _V )
34	33	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> A e. _V )
35	32	brrelex1i 4769	. . . . . . . 8 |- ( C ~<_ D -> C e. _V )
36	35	ad3antlr 493	. . . . . . 7 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> C e. _V )
37		djuex 7241	. . . . . . 7 |- ( ( A e. _V /\ C e. _V ) -> ( A ⊔ C ) e. _V )
38	34, 36, 37	syl2anc 411	. . . . . 6 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ( A ⊔ C ) e. _V )
39		fex 5882	. . . . . 6 |- ( (case ( ( (inl |` B ) o. f ) , ( (inr |` D ) o. g ) ) : ( A ⊔ C ) --> ( B ⊔ D ) /\ ( A ⊔ C ) e. _V ) -> case ( ( (inl |` B ) o. f ) , ( (inr |` D ) o. g ) ) e. _V )
40	31, 38, 39	syl2anc 411	. . . . 5 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> case ( ( (inl |` B ) o. f ) , ( (inr |` D ) o. g ) ) e. _V )
41		f1eq1 5537	. . . . . 6 |- ( h = case ( ( (inl |` B ) o. f ) , ( (inr |` D ) o. g ) ) -> ( h : ( A ⊔ C ) -1-1-> ( B ⊔ D ) <-> case ( ( (inl |` B ) o. f ) , ( (inr |` D ) o. g ) ) : ( A ⊔ C ) -1-1-> ( B ⊔ D ) ) )
42	41	spcegv 2894	. . . . 5 |- (case ( ( (inl |` B ) o. f ) , ( (inr |` D ) o. g ) ) e. _V -> (case ( ( (inl |` B ) o. f ) , ( (inr |` D ) o. g ) ) : ( A ⊔ C ) -1-1-> ( B ⊔ D ) -> E. h h : ( A ⊔ C ) -1-1-> ( B ⊔ D ) ) )
43	40, 29, 42	sylc 62	. . . 4 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> E. h h : ( A ⊔ C ) -1-1-> ( B ⊔ D ) )
44	32	brrelex2i 4770	. . . . . 6 |- ( A ~<_ B -> B e. _V )
45	44	ad3antrrr 492	. . . . 5 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> B e. _V )
46	32	brrelex2i 4770	. . . . . 6 |- ( C ~<_ D -> D e. _V )
47	46	ad3antlr 493	. . . . 5 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> D e. _V )
48		djuex 7241	. . . . . 6 |- ( ( B e. _V /\ D e. _V ) -> ( B ⊔ D ) e. _V )
49		brdomg 6918	. . . . . 6 |- ( ( B ⊔ D ) e. _V -> ( ( A ⊔ C ) ~<_ ( B ⊔ D ) <-> E. h h : ( A ⊔ C ) -1-1-> ( B ⊔ D ) ) )
50	48, 49	syl 14	. . . . 5 |- ( ( B e. _V /\ D e. _V ) -> ( ( A ⊔ C ) ~<_ ( B ⊔ D ) <-> E. h h : ( A ⊔ C ) -1-1-> ( B ⊔ D ) ) )
51	45, 47, 50	syl2anc 411	. . . 4 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ( ( A ⊔ C ) ~<_ ( B ⊔ D ) <-> E. h h : ( A ⊔ C ) -1-1-> ( B ⊔ D ) ) )
52	43, 51	mpbird 167	. . 3 |- ( ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) /\ g : C -1-1-> D ) -> ( A ⊔ C ) ~<_ ( B ⊔ D ) )
53	4, 52	exlimddv 1947	. 2 |- ( ( ( A ~<_ B /\ C ~<_ D ) /\ f : A -1-1-> B ) -> ( A ⊔ C ) ~<_ ( B ⊔ D ) )
54	2, 53	exlimddv 1947	1 |- ( ( A ~<_ B /\ C ~<_ D ) -> ( A ⊔ C ) ~<_ ( B ⊔ D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   i^i cin 3199   C_ wss 3200   (/)c0 3494   class class class wbr 4088   ran crn 4726   |` cres 4727   o. ccom 4729   -->wf 5322   -1-1->wf1 5323   ~<_ cdom 6907   ⊔ cdju 7235  inlcinl 7243  inrcinr 7244  casecdjucase 7281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-dom 6910  df-dju 7236  df-inl 7245  df-inr 7246  df-case 7282

This theorem is referenced by:  exmidfodomrlemr  7412  exmidfodomrlemrALT  7413  sbthom  16630