Theorem fodomr 7665

Description: There exists a mapping from a set onto any (nonempty) set that it dominates. (Contributed by NM, 23-Mar-2006.)

Assertion

Ref	Expression
fodomr	|- ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B )

Distinct variable groups:   A, f   B, f

Proof of Theorem fodomr

Dummy variables g z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 7519	. . . 4 |- Rel ~<_
2	1	brrelex2i 5040	. . 3 |- ( B ~<_ A -> A e. _V )
3	2	adantl 466	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> A e. _V )
4	1	brrelexi 5039	. . . 4 |- ( B ~<_ A -> B e. _V )
5		0sdomg 7643	. . . . 5 |- ( B e. _V -> ( (/) ~< B <-> B =/= (/) ) )
6		n0 3794	. . . . 5 |- ( B =/= (/) <-> E. z z e. B )
7	5, 6	syl6bb 261	. . . 4 |- ( B e. _V -> ( (/) ~< B <-> E. z z e. B ) )
8	4, 7	syl 16	. . 3 |- ( B ~<_ A -> ( (/) ~< B <-> E. z z e. B ) )
9	8	biimpac 486	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. z z e. B )
10		brdomi 7524	. . 3 |- ( B ~<_ A -> E. g g : B -1-1-> A )
11	10	adantl 466	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. g g : B -1-1-> A )
12		difexg 4595	. . . . . . . . . 10 |- ( A e. _V -> ( A \ ran g ) e. _V )
13		snex 4688	. . . . . . . . . 10 |- { z } e. _V
14		xpexg 6709	. . . . . . . . . 10 |- ( ( ( A \ ran g ) e. _V /\ { z } e. _V ) -> ( ( A \ ran g ) X. { z } ) e. _V )
15	12, 13, 14	sylancl 662	. . . . . . . . 9 |- ( A e. _V -> ( ( A \ ran g ) X. { z } ) e. _V )
16		vex 3116	. . . . . . . . . 10 |- g e. _V
17	16	cnvex 6728	. . . . . . . . 9 |- `' g e. _V
18	15, 17	jctil 537	. . . . . . . 8 |- ( A e. _V -> ( `' g e. _V /\ ( ( A \ ran g ) X. { z } ) e. _V ) )
19		unexb 6582	. . . . . . . 8 |- ( ( `' g e. _V /\ ( ( A \ ran g ) X. { z } ) e. _V ) <-> ( `' g u. ( ( A \ ran g ) X. { z } ) ) e. _V )
20	18, 19	sylib 196	. . . . . . 7 |- ( A e. _V -> ( `' g u. ( ( A \ ran g ) X. { z } ) ) e. _V )
21		df-f1 5591	. . . . . . . . . . . . 13 |- ( g : B -1-1-> A <-> ( g : B --> A /\ Fun `' g ) )
22	21	simprbi 464	. . . . . . . . . . . 12 |- ( g : B -1-1-> A -> Fun `' g )
23		vex 3116	. . . . . . . . . . . . . 14 |- z e. _V
24	23	fconst 5769	. . . . . . . . . . . . 13 |- ( ( A \ ran g ) X. { z } ) : ( A \ ran g ) --> { z }
25		ffun 5731	. . . . . . . . . . . . 13 |- ( ( ( A \ ran g ) X. { z } ) : ( A \ ran g ) --> { z } -> Fun ( ( A \ ran g ) X. { z } ) )
26	24, 25	ax-mp 5	. . . . . . . . . . . 12 |- Fun ( ( A \ ran g ) X. { z } )
27	22, 26	jctir 538	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ( Fun `' g /\ Fun ( ( A \ ran g ) X. { z } ) ) )
28		df-rn 5010	. . . . . . . . . . . . . 14 |- ran g = dom `' g
29	28	eqcomi 2480	. . . . . . . . . . . . 13 |- dom `' g = ran g
30	23	snnz 4145	. . . . . . . . . . . . . 14 |- { z } =/= (/)
31		dmxp 5219	. . . . . . . . . . . . . 14 |- ( { z } =/= (/) -> dom ( ( A \ ran g ) X. { z } ) = ( A \ ran g ) )
32	30, 31	ax-mp 5	. . . . . . . . . . . . 13 |- dom ( ( A \ ran g ) X. { z } ) = ( A \ ran g )
33	29, 32	ineq12i 3698	. . . . . . . . . . . 12 |- ( dom `' g i^i dom ( ( A \ ran g ) X. { z } ) ) = ( ran g i^i ( A \ ran g ) )
34		disjdif 3899	. . . . . . . . . . . 12 |- ( ran g i^i ( A \ ran g ) ) = (/)
35	33, 34	eqtri 2496	. . . . . . . . . . 11 |- ( dom `' g i^i dom ( ( A \ ran g ) X. { z } ) ) = (/)
36		funun 5628	. . . . . . . . . . 11 |- ( ( ( Fun `' g /\ Fun ( ( A \ ran g ) X. { z } ) ) /\ ( dom `' g i^i dom ( ( A \ ran g ) X. { z } ) ) = (/) ) -> Fun ( `' g u. ( ( A \ ran g ) X. { z } ) ) )
37	27, 35, 36	sylancl 662	. . . . . . . . . 10 |- ( g : B -1-1-> A -> Fun ( `' g u. ( ( A \ ran g ) X. { z } ) ) )
38	37	adantl 466	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> Fun ( `' g u. ( ( A \ ran g ) X. { z } ) ) )
39		dmun 5207	. . . . . . . . . . . 12 |- dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( dom `' g u. dom ( ( A \ ran g ) X. { z } ) )
40	28	uneq1i 3654	. . . . . . . . . . . 12 |- ( ran g u. dom ( ( A \ ran g ) X. { z } ) ) = ( dom `' g u. dom ( ( A \ ran g ) X. { z } ) )
41	32	uneq2i 3655	. . . . . . . . . . . 12 |- ( ran g u. dom ( ( A \ ran g ) X. { z } ) ) = ( ran g u. ( A \ ran g ) )
42	39, 40, 41	3eqtr2i 2502	. . . . . . . . . . 11 |- dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( ran g u. ( A \ ran g ) )
43		f1f 5779	. . . . . . . . . . . . 13 |- ( g : B -1-1-> A -> g : B --> A )
44		frn 5735	. . . . . . . . . . . . 13 |- ( g : B --> A -> ran g C_ A )
45	43, 44	syl 16	. . . . . . . . . . . 12 |- ( g : B -1-1-> A -> ran g C_ A )
46		undif 3907	. . . . . . . . . . . 12 |- ( ran g C_ A <-> ( ran g u. ( A \ ran g ) ) = A )
47	45, 46	sylib 196	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ( ran g u. ( A \ ran g ) ) = A )
48	42, 47	syl5eq 2520	. . . . . . . . . 10 |- ( g : B -1-1-> A -> dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = A )
49	48	adantl 466	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = A )
50		df-fn 5589	. . . . . . . . 9 |- ( ( `' g u. ( ( A \ ran g ) X. { z } ) ) Fn A <-> ( Fun ( `' g u. ( ( A \ ran g ) X. { z } ) ) /\ dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = A ) )
51	38, 49, 50	sylanbrc 664	. . . . . . . 8 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( `' g u. ( ( A \ ran g ) X. { z } ) ) Fn A )
52		rnun 5412	. . . . . . . . 9 |- ran ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( ran `' g u. ran ( ( A \ ran g ) X. { z } ) )
53		dfdm4 5193	. . . . . . . . . . . 12 |- dom g = ran `' g
54		f1dm 5783	. . . . . . . . . . . 12 |- ( g : B -1-1-> A -> dom g = B )
55	53, 54	syl5eqr 2522	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ran `' g = B )
56	55	uneq1d 3657	. . . . . . . . . 10 |- ( g : B -1-1-> A -> ( ran `' g u. ran ( ( A \ ran g ) X. { z } ) ) = ( B u. ran ( ( A \ ran g ) X. { z } ) ) )
57		xpeq1 5013	. . . . . . . . . . . . . . . . 17 |- ( ( A \ ran g ) = (/) -> ( ( A \ ran g ) X. { z } ) = ( (/) X. { z } ) )
58		0xp 5078	. . . . . . . . . . . . . . . . 17 |- ( (/) X. { z } ) = (/)
59	57, 58	syl6eq 2524	. . . . . . . . . . . . . . . 16 |- ( ( A \ ran g ) = (/) -> ( ( A \ ran g ) X. { z } ) = (/) )
60	59	rneqd 5228	. . . . . . . . . . . . . . 15 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) = ran (/) )
61		rn0 5252	. . . . . . . . . . . . . . 15 |- ran (/) = (/)
62	60, 61	syl6eq 2524	. . . . . . . . . . . . . 14 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) = (/) )
63		0ss 3814	. . . . . . . . . . . . . 14 |- (/) C_ B
64	62, 63	syl6eqss 3554	. . . . . . . . . . . . 13 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) C_ B )
65	64	a1d 25	. . . . . . . . . . . 12 |- ( ( A \ ran g ) = (/) -> ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B ) )
66		rnxp 5435	. . . . . . . . . . . . . . 15 |- ( ( A \ ran g ) =/= (/) -> ran ( ( A \ ran g ) X. { z } ) = { z } )
67	66	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> ran ( ( A \ ran g ) X. { z } ) = { z } )
68		snssi 4171	. . . . . . . . . . . . . . 15 |- ( z e. B -> { z } C_ B )
69	68	adantl 466	. . . . . . . . . . . . . 14 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> { z } C_ B )
70	67, 69	eqsstrd 3538	. . . . . . . . . . . . 13 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> ran ( ( A \ ran g ) X. { z } ) C_ B )
71	70	ex 434	. . . . . . . . . . . 12 |- ( ( A \ ran g ) =/= (/) -> ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B ) )
72	65, 71	pm2.61ine 2780	. . . . . . . . . . 11 |- ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B )
73		ssequn2 3677	. . . . . . . . . . 11 |- ( ran ( ( A \ ran g ) X. { z } ) C_ B <-> ( B u. ran ( ( A \ ran g ) X. { z } ) ) = B )
74	72, 73	sylib 196	. . . . . . . . . 10 |- ( z e. B -> ( B u. ran ( ( A \ ran g ) X. { z } ) ) = B )
75	56, 74	sylan9eqr 2530	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( ran `' g u. ran ( ( A \ ran g ) X. { z } ) ) = B )
76	52, 75	syl5eq 2520	. . . . . . . 8 |- ( ( z e. B /\ g : B -1-1-> A ) -> ran ( `' g u. ( ( A \ ran g ) X. { z } ) ) = B )
77		df-fo 5592	. . . . . . . 8 |- ( ( `' g u. ( ( A \ ran g ) X. { z } ) ) : A -onto-> B <-> ( ( `' g u. ( ( A \ ran g ) X. { z } ) ) Fn A /\ ran ( `' g u. ( ( A \ ran g ) X. { z } ) ) = B ) )
78	51, 76, 77	sylanbrc 664	. . . . . . 7 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( `' g u. ( ( A \ ran g ) X. { z } ) ) : A -onto-> B )
79		foeq1 5789	. . . . . . . 8 |- ( f = ( `' g u. ( ( A \ ran g ) X. { z } ) ) -> ( f : A -onto-> B <-> ( `' g u. ( ( A \ ran g ) X. { z } ) ) : A -onto-> B ) )
80	79	spcegv 3199	. . . . . . 7 |- ( ( `' g u. ( ( A \ ran g ) X. { z } ) ) e. _V -> ( ( `' g u. ( ( A \ ran g ) X. { z } ) ) : A -onto-> B -> E. f f : A -onto-> B ) )
81	20, 78, 80	syl2im 38	. . . . . 6 |- ( A e. _V -> ( ( z e. B /\ g : B -1-1-> A ) -> E. f f : A -onto-> B ) )
82	81	expdimp 437	. . . . 5 |- ( ( A e. _V /\ z e. B ) -> ( g : B -1-1-> A -> E. f f : A -onto-> B ) )
83	82	exlimdv 1700	. . . 4 |- ( ( A e. _V /\ z e. B ) -> ( E. g g : B -1-1-> A -> E. f f : A -onto-> B ) )
84	83	ex 434	. . 3 |- ( A e. _V -> ( z e. B -> ( E. g g : B -1-1-> A -> E. f f : A -onto-> B ) ) )
85	84	exlimdv 1700	. 2 |- ( A e. _V -> ( E. z z e. B -> ( E. g g : B -1-1-> A -> E. f f : A -onto-> B ) ) )
86	3, 9, 11, 85	syl3c 61	1 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   =/= wne 2662   _Vcvv 3113   \ cdif 3473   u. cun 3474   i^i cin 3475   C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447   X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   Fun wfun 5580   Fn wfn 5581   -->wf 5582   -1-1->wf1 5583   -onto->wfo 5584   ~<_ cdom 7511   ~< csdm 7512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516

This theorem is referenced by:  pwdom  7666  fodomfib  7796  domwdom  7996  iunfictbso  8491  fodomb  8900  brdom3  8902  konigthlem  8939  1stcfb  19712  ovoliunnul  21653  ovoliunnfl  29633  voliunnfl  29635  volsupnfl  29636