Theorem fodomr 7217

Description: There exists a mapping from a set onto any (non-empty) 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 7074	. . . 4 |- Rel ~<_
2	1	brrelex2i 4878	. . 3 |- ( B ~<_ A -> A e. _V )
3	2	adantl 453	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> A e. _V )
4	1	brrelexi 4877	. . . 4 |- ( B ~<_ A -> B e. _V )
5		0sdomg 7195	. . . . 5 |- ( B e. _V -> ( (/) ~< B <-> B =/= (/) ) )
6		n0 3597	. . . . 5 |- ( B =/= (/) <-> E. z z e. B )
7	5, 6	syl6bb 253	. . . 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 473	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. z z e. B )
10		brdomi 7078	. . 3 |- ( B ~<_ A -> E. g g : B -1-1-> A )
11	10	adantl 453	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. g g : B -1-1-> A )
12		difexg 4311	. . . . . . . . . 10 |- ( A e. _V -> ( A \ ran g ) e. _V )
13		snex 4365	. . . . . . . . . 10 |- { z } e. _V
14		xpexg 4948	. . . . . . . . . 10 |- ( ( ( A \ ran g ) e. _V /\ { z } e. _V ) -> ( ( A \ ran g ) X. { z } ) e. _V )
15	12, 13, 14	sylancl 644	. . . . . . . . 9 |- ( A e. _V -> ( ( A \ ran g ) X. { z } ) e. _V )
16		vex 2919	. . . . . . . . . 10 |- g e. _V
17	16	cnvex 5365	. . . . . . . . 9 |- `' g e. _V
18	15, 17	jctil 524	. . . . . . . 8 |- ( A e. _V -> ( `' g e. _V /\ ( ( A \ ran g ) X. { z } ) e. _V ) )
19		unexb 4668	. . . . . . . 8 |- ( ( `' g e. _V /\ ( ( A \ ran g ) X. { z } ) e. _V ) <-> ( `' g u. ( ( A \ ran g ) X. { z } ) ) e. _V )
20	18, 19	sylib 189	. . . . . . 7 |- ( A e. _V -> ( `' g u. ( ( A \ ran g ) X. { z } ) ) e. _V )
21		df-f1 5418	. . . . . . . . . . . . 13 |- ( g : B -1-1-> A <-> ( g : B --> A /\ Fun `' g ) )
22	21	simprbi 451	. . . . . . . . . . . 12 |- ( g : B -1-1-> A -> Fun `' g )
23		vex 2919	. . . . . . . . . . . . . 14 |- z e. _V
24	23	fconst 5588	. . . . . . . . . . . . 13 |- ( ( A \ ran g ) X. { z } ) : ( A \ ran g ) --> { z }
25		ffun 5552	. . . . . . . . . . . . 13 |- ( ( ( A \ ran g ) X. { z } ) : ( A \ ran g ) --> { z } -> Fun ( ( A \ ran g ) X. { z } ) )
26	24, 25	ax-mp 8	. . . . . . . . . . . 12 |- Fun ( ( A \ ran g ) X. { z } )
27	22, 26	jctir 525	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ( Fun `' g /\ Fun ( ( A \ ran g ) X. { z } ) ) )
28		df-rn 4848	. . . . . . . . . . . . . 14 |- ran g = dom `' g
29	28	eqcomi 2408	. . . . . . . . . . . . 13 |- dom `' g = ran g
30	23	snnz 3882	. . . . . . . . . . . . . 14 |- { z } =/= (/)
31		dmxp 5047	. . . . . . . . . . . . . 14 |- ( { z } =/= (/) -> dom ( ( A \ ran g ) X. { z } ) = ( A \ ran g ) )
32	30, 31	ax-mp 8	. . . . . . . . . . . . 13 |- dom ( ( A \ ran g ) X. { z } ) = ( A \ ran g )
33	29, 32	ineq12i 3500	. . . . . . . . . . . 12 |- ( dom `' g i^i dom ( ( A \ ran g ) X. { z } ) ) = ( ran g i^i ( A \ ran g ) )
34		disjdif 3660	. . . . . . . . . . . 12 |- ( ran g i^i ( A \ ran g ) ) = (/)
35	33, 34	eqtri 2424	. . . . . . . . . . 11 |- ( dom `' g i^i dom ( ( A \ ran g ) X. { z } ) ) = (/)
36		funun 5454	. . . . . . . . . . 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 644	. . . . . . . . . 10 |- ( g : B -1-1-> A -> Fun ( `' g u. ( ( A \ ran g ) X. { z } ) ) )
38	37	adantl 453	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> Fun ( `' g u. ( ( A \ ran g ) X. { z } ) ) )
39		dmun 5035	. . . . . . . . . . . 12 |- dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( dom `' g u. dom ( ( A \ ran g ) X. { z } ) )
40	28	uneq1i 3457	. . . . . . . . . . . 12 |- ( ran g u. dom ( ( A \ ran g ) X. { z } ) ) = ( dom `' g u. dom ( ( A \ ran g ) X. { z } ) )
41	32	uneq2i 3458	. . . . . . . . . . . 12 |- ( ran g u. dom ( ( A \ ran g ) X. { z } ) ) = ( ran g u. ( A \ ran g ) )
42	39, 40, 41	3eqtr2i 2430	. . . . . . . . . . 11 |- dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( ran g u. ( A \ ran g ) )
43		f1f 5598	. . . . . . . . . . . . 13 |- ( g : B -1-1-> A -> g : B --> A )
44		frn 5556	. . . . . . . . . . . . 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 3668	. . . . . . . . . . . 12 |- ( ran g C_ A <-> ( ran g u. ( A \ ran g ) ) = A )
47	45, 46	sylib 189	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ( ran g u. ( A \ ran g ) ) = A )
48	42, 47	syl5eq 2448	. . . . . . . . . 10 |- ( g : B -1-1-> A -> dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = A )
49	48	adantl 453	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = A )
50		df-fn 5416	. . . . . . . . 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 646	. . . . . . . 8 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( `' g u. ( ( A \ ran g ) X. { z } ) ) Fn A )
52		rnun 5239	. . . . . . . . 9 |- ran ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( ran `' g u. ran ( ( A \ ran g ) X. { z } ) )
53		dfdm4 5022	. . . . . . . . . . . 12 |- dom g = ran `' g
54		f1dm 5602	. . . . . . . . . . . 12 |- ( g : B -1-1-> A -> dom g = B )
55	53, 54	syl5eqr 2450	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ran `' g = B )
56	55	uneq1d 3460	. . . . . . . . . 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 4851	. . . . . . . . . . . . . . . . 17 |- ( ( A \ ran g ) = (/) -> ( ( A \ ran g ) X. { z } ) = ( (/) X. { z } ) )
58		xp0r 4915	. . . . . . . . . . . . . . . . 17 |- ( (/) X. { z } ) = (/)
59	57, 58	syl6eq 2452	. . . . . . . . . . . . . . . 16 |- ( ( A \ ran g ) = (/) -> ( ( A \ ran g ) X. { z } ) = (/) )
60	59	rneqd 5056	. . . . . . . . . . . . . . 15 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) = ran (/) )
61		rn0 5086	. . . . . . . . . . . . . . 15 |- ran (/) = (/)
62	60, 61	syl6eq 2452	. . . . . . . . . . . . . 14 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) = (/) )
63		0ss 3616	. . . . . . . . . . . . . 14 |- (/) C_ B
64	62, 63	syl6eqss 3358	. . . . . . . . . . . . 13 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) C_ B )
65	64	a1d 23	. . . . . . . . . . . 12 |- ( ( A \ ran g ) = (/) -> ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B ) )
66		rnxp 5258	. . . . . . . . . . . . . . 15 |- ( ( A \ ran g ) =/= (/) -> ran ( ( A \ ran g ) X. { z } ) = { z } )
67	66	adantr 452	. . . . . . . . . . . . . 14 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> ran ( ( A \ ran g ) X. { z } ) = { z } )
68		snssi 3902	. . . . . . . . . . . . . . 15 |- ( z e. B -> { z } C_ B )
69	68	adantl 453	. . . . . . . . . . . . . 14 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> { z } C_ B )
70	67, 69	eqsstrd 3342	. . . . . . . . . . . . 13 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> ran ( ( A \ ran g ) X. { z } ) C_ B )
71	70	ex 424	. . . . . . . . . . . 12 |- ( ( A \ ran g ) =/= (/) -> ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B ) )
72	65, 71	pm2.61ine 2643	. . . . . . . . . . 11 |- ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B )
73		ssequn2 3480	. . . . . . . . . . 11 |- ( ran ( ( A \ ran g ) X. { z } ) C_ B <-> ( B u. ran ( ( A \ ran g ) X. { z } ) ) = B )
74	72, 73	sylib 189	. . . . . . . . . 10 |- ( z e. B -> ( B u. ran ( ( A \ ran g ) X. { z } ) ) = B )
75	56, 74	sylan9eqr 2458	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( ran `' g u. ran ( ( A \ ran g ) X. { z } ) ) = B )
76	52, 75	syl5eq 2448	. . . . . . . 8 |- ( ( z e. B /\ g : B -1-1-> A ) -> ran ( `' g u. ( ( A \ ran g ) X. { z } ) ) = B )
77		df-fo 5419	. . . . . . . 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 646	. . . . . . 7 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( `' g u. ( ( A \ ran g ) X. { z } ) ) : A -onto-> B )
79		foeq1 5608	. . . . . . . 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 2997	. . . . . . 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 36	. . . . . 6 |- ( A e. _V -> ( ( z e. B /\ g : B -1-1-> A ) -> E. f f : A -onto-> B ) )
82	81	expdimp 427	. . . . 5 |- ( ( A e. _V /\ z e. B ) -> ( g : B -1-1-> A -> E. f f : A -onto-> B ) )
83	82	exlimdv 1643	. . . 4 |- ( ( A e. _V /\ z e. B ) -> ( E. g g : B -1-1-> A -> E. f f : A -onto-> B ) )
84	83	ex 424	. . 3 |- ( A e. _V -> ( z e. B -> ( E. g g : B -1-1-> A -> E. f f : A -onto-> B ) ) )
85	84	exlimdv 1643	. 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 59	1 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   E.wex 1547   = wceq 1649   e. wcel 1721   =/= wne 2567   _Vcvv 2916   \ cdif 3277   u. cun 3278   i^i cin 3279   C_ wss 3280   (/)c0 3588   {csn 3774   class class class wbr 4172   X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838   Fun wfun 5407   Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -onto->wfo 5411   ~<_ cdom 7066   ~< csdm 7067

This theorem is referenced by:  pwdom  7218  fodomfib  7345  domwdom  7498  iunfictbso  7951  fodomb  8360  brdom3  8362  konigthlem  8399  1stcfb  17461  ovoliunnul  19356  ovoliunnfl  26147  voliunnfl  26149  volsupnfl  26150

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071