Theorem fodomr 7667

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 7521	. . . 4 |- Rel ~<_
2	1	brrelex2i 5028	. . 3 |- ( B ~<_ A -> A e. _V )
3	2	adantl 466	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> A e. _V )
4	1	brrelexi 5027	. . . 4 |- ( B ~<_ A -> B e. _V )
5		0sdomg 7645	. . . . 5 |- ( B e. _V -> ( (/) ~< B <-> B =/= (/) ) )
6		n0 3777	. . . . 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 7526	. . 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 4582	. . . . . . . . . 10 |- ( A e. _V -> ( A \ ran g ) e. _V )
13		snex 4675	. . . . . . . . . 10 |- { z } e. _V
14		xpexg 6584	. . . . . . . . . 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 3096	. . . . . . . . . 10 |- g e. _V
17	16	cnvex 6729	. . . . . . . . 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 5580	. . . . . . . . . . . . 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 3096	. . . . . . . . . . . . . 14 |- z e. _V
24	23	fconst 5758	. . . . . . . . . . . . 13 |- ( ( A \ ran g ) X. { z } ) : ( A \ ran g ) --> { z }
25		ffun 5720	. . . . . . . . . . . . 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 4997	. . . . . . . . . . . . . 14 |- ran g = dom `' g
29	28	eqcomi 2454	. . . . . . . . . . . . 13 |- dom `' g = ran g
30	23	snnz 4130	. . . . . . . . . . . . . 14 |- { z } =/= (/)
31		dmxp 5208	. . . . . . . . . . . . . 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 3681	. . . . . . . . . . . 12 |- ( dom `' g i^i dom ( ( A \ ran g ) X. { z } ) ) = ( ran g i^i ( A \ ran g ) )
34		disjdif 3883	. . . . . . . . . . . 12 |- ( ran g i^i ( A \ ran g ) ) = (/)
35	33, 34	eqtri 2470	. . . . . . . . . . 11 |- ( dom `' g i^i dom ( ( A \ ran g ) X. { z } ) ) = (/)
36		funun 5617	. . . . . . . . . . 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 5196	. . . . . . . . . . . 12 |- dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( dom `' g u. dom ( ( A \ ran g ) X. { z } ) )
40	28	uneq1i 3637	. . . . . . . . . . . 12 |- ( ran g u. dom ( ( A \ ran g ) X. { z } ) ) = ( dom `' g u. dom ( ( A \ ran g ) X. { z } ) )
41	32	uneq2i 3638	. . . . . . . . . . . 12 |- ( ran g u. dom ( ( A \ ran g ) X. { z } ) ) = ( ran g u. ( A \ ran g ) )
42	39, 40, 41	3eqtr2i 2476	. . . . . . . . . . 11 |- dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( ran g u. ( A \ ran g ) )
43		f1f 5768	. . . . . . . . . . . . 13 |- ( g : B -1-1-> A -> g : B --> A )
44		frn 5724	. . . . . . . . . . . . 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 3891	. . . . . . . . . . . 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 2494	. . . . . . . . . 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 5578	. . . . . . . . 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 5401	. . . . . . . . 9 |- ran ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( ran `' g u. ran ( ( A \ ran g ) X. { z } ) )
53		dfdm4 5182	. . . . . . . . . . . 12 |- dom g = ran `' g
54		f1dm 5772	. . . . . . . . . . . 12 |- ( g : B -1-1-> A -> dom g = B )
55	53, 54	syl5eqr 2496	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ran `' g = B )
56	55	uneq1d 3640	. . . . . . . . . 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 5000	. . . . . . . . . . . . . . . . 17 |- ( ( A \ ran g ) = (/) -> ( ( A \ ran g ) X. { z } ) = ( (/) X. { z } ) )
58		0xp 5067	. . . . . . . . . . . . . . . . 17 |- ( (/) X. { z } ) = (/)
59	57, 58	syl6eq 2498	. . . . . . . . . . . . . . . 16 |- ( ( A \ ran g ) = (/) -> ( ( A \ ran g ) X. { z } ) = (/) )
60	59	rneqd 5217	. . . . . . . . . . . . . . 15 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) = ran (/) )
61		rn0 5241	. . . . . . . . . . . . . . 15 |- ran (/) = (/)
62	60, 61	syl6eq 2498	. . . . . . . . . . . . . 14 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) = (/) )
63		0ss 3797	. . . . . . . . . . . . . 14 |- (/) C_ B
64	62, 63	syl6eqss 3537	. . . . . . . . . . . . 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 5424	. . . . . . . . . . . . . . 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 4156	. . . . . . . . . . . . . . 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 3521	. . . . . . . . . . . . 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 2754	. . . . . . . . . . 11 |- ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B )
73		ssequn2 3660	. . . . . . . . . . 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 2504	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( ran `' g u. ran ( ( A \ ran g ) X. { z } ) ) = B )
76	52, 75	syl5eq 2494	. . . . . . . 8 |- ( ( z e. B /\ g : B -1-1-> A ) -> ran ( `' g u. ( ( A \ ran g ) X. { z } ) ) = B )
77		df-fo 5581	. . . . . . . 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 5778	. . . . . . . 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 3179	. . . . . . 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 1709	. . . 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 1709	. 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 1381   E.wex 1597   e. wcel 1802   =/= wne 2636   _Vcvv 3093   \ cdif 3456   u. cun 3457   i^i cin 3458   C_ wss 3459   (/)c0 3768   {csn 4011   class class class wbr 4434   X. cxp 4984   `'ccnv 4985   dom cdm 4986   ran crn 4987   Fun wfun 5569   Fn wfn 5570   -->wf 5571   -1-1->wf1 5572   -onto->wfo 5573   ~<_ cdom 7513   ~< csdm 7514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518

This theorem is referenced by:  pwdom  7668  fodomfib  7799  domwdom  8000  iunfictbso  8495  fodomb  8904  brdom3  8906  konigthlem  8943  1stcfb  19816  ovoliunnul  21788  ovoliunnfl  30028  voliunnfl  30030  volsupnfl  30031