Theorem fodomr 7605

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 7459	. . . 4 |- Rel ~<_
2	1	brrelex2i 4968	. . 3 |- ( B ~<_ A -> A e. _V )
3	2	adantl 464	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> A e. _V )
4	1	brrelexi 4967	. . . 4 |- ( B ~<_ A -> B e. _V )
5		0sdomg 7583	. . . . 5 |- ( B e. _V -> ( (/) ~< B <-> B =/= (/) ) )
6		n0 3734	. . . . 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 484	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. z z e. B )
10		brdomi 7464	. . 3 |- ( B ~<_ A -> E. g g : B -1-1-> A )
11	10	adantl 464	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. g g : B -1-1-> A )
12		difexg 4526	. . . . . . . . . 10 |- ( A e. _V -> ( A \ ran g ) e. _V )
13		snex 4616	. . . . . . . . . 10 |- { z } e. _V
14		xpexg 6519	. . . . . . . . . 10 |- ( ( ( A \ ran g ) e. _V /\ { z } e. _V ) -> ( ( A \ ran g ) X. { z } ) e. _V )
15	12, 13, 14	sylancl 660	. . . . . . . . 9 |- ( A e. _V -> ( ( A \ ran g ) X. { z } ) e. _V )
16		vex 3050	. . . . . . . . . 10 |- g e. _V
17	16	cnvex 6664	. . . . . . . . 9 |- `' g e. _V
18	15, 17	jctil 535	. . . . . . . 8 |- ( A e. _V -> ( `' g e. _V /\ ( ( A \ ran g ) X. { z } ) e. _V ) )
19		unexb 6517	. . . . . . . 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 5514	. . . . . . . . . . . . 13 |- ( g : B -1-1-> A <-> ( g : B --> A /\ Fun `' g ) )
22	21	simprbi 462	. . . . . . . . . . . 12 |- ( g : B -1-1-> A -> Fun `' g )
23		vex 3050	. . . . . . . . . . . . . 14 |- z e. _V
24	23	fconst 5692	. . . . . . . . . . . . 13 |- ( ( A \ ran g ) X. { z } ) : ( A \ ran g ) --> { z }
25		ffun 5654	. . . . . . . . . . . . 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 536	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ( Fun `' g /\ Fun ( ( A \ ran g ) X. { z } ) ) )
28		df-rn 4937	. . . . . . . . . . . . . 14 |- ran g = dom `' g
29	28	eqcomi 2405	. . . . . . . . . . . . 13 |- dom `' g = ran g
30	23	snnz 4075	. . . . . . . . . . . . . 14 |- { z } =/= (/)
31		dmxp 5147	. . . . . . . . . . . . . 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 3625	. . . . . . . . . . . 12 |- ( dom `' g i^i dom ( ( A \ ran g ) X. { z } ) ) = ( ran g i^i ( A \ ran g ) )
34		disjdif 3829	. . . . . . . . . . . 12 |- ( ran g i^i ( A \ ran g ) ) = (/)
35	33, 34	eqtri 2421	. . . . . . . . . . 11 |- ( dom `' g i^i dom ( ( A \ ran g ) X. { z } ) ) = (/)
36		funun 5551	. . . . . . . . . . 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 660	. . . . . . . . . 10 |- ( g : B -1-1-> A -> Fun ( `' g u. ( ( A \ ran g ) X. { z } ) ) )
38	37	adantl 464	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> Fun ( `' g u. ( ( A \ ran g ) X. { z } ) ) )
39		dmun 5135	. . . . . . . . . . . 12 |- dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( dom `' g u. dom ( ( A \ ran g ) X. { z } ) )
40	28	uneq1i 3581	. . . . . . . . . . . 12 |- ( ran g u. dom ( ( A \ ran g ) X. { z } ) ) = ( dom `' g u. dom ( ( A \ ran g ) X. { z } ) )
41	32	uneq2i 3582	. . . . . . . . . . . 12 |- ( ran g u. dom ( ( A \ ran g ) X. { z } ) ) = ( ran g u. ( A \ ran g ) )
42	39, 40, 41	3eqtr2i 2427	. . . . . . . . . . 11 |- dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( ran g u. ( A \ ran g ) )
43		f1f 5702	. . . . . . . . . . . . 13 |- ( g : B -1-1-> A -> g : B --> A )
44		frn 5658	. . . . . . . . . . . . 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 3837	. . . . . . . . . . . 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 2445	. . . . . . . . . 10 |- ( g : B -1-1-> A -> dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = A )
49	48	adantl 464	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = A )
50		df-fn 5512	. . . . . . . . 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 662	. . . . . . . 8 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( `' g u. ( ( A \ ran g ) X. { z } ) ) Fn A )
52		rnun 5337	. . . . . . . . 9 |- ran ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( ran `' g u. ran ( ( A \ ran g ) X. { z } ) )
53		dfdm4 5121	. . . . . . . . . . . 12 |- dom g = ran `' g
54		f1dm 5706	. . . . . . . . . . . 12 |- ( g : B -1-1-> A -> dom g = B )
55	53, 54	syl5eqr 2447	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ran `' g = B )
56	55	uneq1d 3584	. . . . . . . . . 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 4940	. . . . . . . . . . . . . . . . 17 |- ( ( A \ ran g ) = (/) -> ( ( A \ ran g ) X. { z } ) = ( (/) X. { z } ) )
58		0xp 5007	. . . . . . . . . . . . . . . . 17 |- ( (/) X. { z } ) = (/)
59	57, 58	syl6eq 2449	. . . . . . . . . . . . . . . 16 |- ( ( A \ ran g ) = (/) -> ( ( A \ ran g ) X. { z } ) = (/) )
60	59	rneqd 5156	. . . . . . . . . . . . . . 15 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) = ran (/) )
61		rn0 5180	. . . . . . . . . . . . . . 15 |- ran (/) = (/)
62	60, 61	syl6eq 2449	. . . . . . . . . . . . . 14 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) = (/) )
63		0ss 3754	. . . . . . . . . . . . . 14 |- (/) C_ B
64	62, 63	syl6eqss 3480	. . . . . . . . . . . . 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 5360	. . . . . . . . . . . . . . 15 |- ( ( A \ ran g ) =/= (/) -> ran ( ( A \ ran g ) X. { z } ) = { z } )
67	66	adantr 463	. . . . . . . . . . . . . 14 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> ran ( ( A \ ran g ) X. { z } ) = { z } )
68		snssi 4101	. . . . . . . . . . . . . . 15 |- ( z e. B -> { z } C_ B )
69	68	adantl 464	. . . . . . . . . . . . . 14 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> { z } C_ B )
70	67, 69	eqsstrd 3464	. . . . . . . . . . . . 13 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> ran ( ( A \ ran g ) X. { z } ) C_ B )
71	70	ex 432	. . . . . . . . . . . 12 |- ( ( A \ ran g ) =/= (/) -> ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B ) )
72	65, 71	pm2.61ine 2705	. . . . . . . . . . 11 |- ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B )
73		ssequn2 3604	. . . . . . . . . . 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 2455	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( ran `' g u. ran ( ( A \ ran g ) X. { z } ) ) = B )
76	52, 75	syl5eq 2445	. . . . . . . 8 |- ( ( z e. B /\ g : B -1-1-> A ) -> ran ( `' g u. ( ( A \ ran g ) X. { z } ) ) = B )
77		df-fo 5515	. . . . . . . 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 662	. . . . . . 7 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( `' g u. ( ( A \ ran g ) X. { z } ) ) : A -onto-> B )
79		foeq1 5712	. . . . . . . 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 3133	. . . . . . 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 435	. . . . 5 |- ( ( A e. _V /\ z e. B ) -> ( g : B -1-1-> A -> E. f f : A -onto-> B ) )
83	82	exlimdv 1739	. . . 4 |- ( ( A e. _V /\ z e. B ) -> ( E. g g : B -1-1-> A -> E. f f : A -onto-> B ) )
84	83	ex 432	. . 3 |- ( A e. _V -> ( z e. B -> ( E. g g : B -1-1-> A -> E. f f : A -onto-> B ) ) )
85	84	exlimdv 1739	. 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 367   = wceq 1399   E.wex 1627   e. wcel 1836   =/= wne 2587   _Vcvv 3047   \ cdif 3399   u. cun 3400   i^i cin 3401   C_ wss 3402   (/)c0 3724   {csn 3957   class class class wbr 4380   X. cxp 4924   `'ccnv 4925   dom cdm 4926   ran crn 4927   Fun wfun 5503   Fn wfn 5504   -->wf 5505   -1-1->wf1 5506   -onto->wfo 5507   ~<_ cdom 7451   ~< csdm 7452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-er 7247  df-en 7454  df-dom 7455  df-sdom 7456

This theorem is referenced by:  pwdom  7606  fodomfib  7733  domwdom  7933  iunfictbso  8426  fodomb  8835  brdom3  8837  konigthlem  8874  1stcfb  20050  ovoliunnul  22022  sigapildsys  28342  carsgclctunlem3  28483  ovoliunnfl  30257  voliunnfl  30259  volsupnfl  30260