Theorem fodomr 7723

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 7575	. . . 4 |- Rel ~<_
2	1	brrelex2i 4876	. . 3 |- ( B ~<_ A -> A e. _V )
3	2	adantl 468	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> A e. _V )
4	1	brrelexi 4875	. . . 4 |- ( B ~<_ A -> B e. _V )
5		0sdomg 7701	. . . . 5 |- ( B e. _V -> ( (/) ~< B <-> B =/= (/) ) )
6		n0 3741	. . . . 5 |- ( B =/= (/) <-> E. z z e. B )
7	5, 6	syl6bb 265	. . . 4 |- ( B e. _V -> ( (/) ~< B <-> E. z z e. B ) )
8	4, 7	syl 17	. . 3 |- ( B ~<_ A -> ( (/) ~< B <-> E. z z e. B ) )
9	8	biimpac 489	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. z z e. B )
10		brdomi 7580	. . 3 |- ( B ~<_ A -> E. g g : B -1-1-> A )
11	10	adantl 468	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. g g : B -1-1-> A )
12		difexg 4551	. . . . . . . . . 10 |- ( A e. _V -> ( A \ ran g ) e. _V )
13		snex 4641	. . . . . . . . . 10 |- { z } e. _V
14		xpexg 6593	. . . . . . . . . 10 |- ( ( ( A \ ran g ) e. _V /\ { z } e. _V ) -> ( ( A \ ran g ) X. { z } ) e. _V )
15	12, 13, 14	sylancl 668	. . . . . . . . 9 |- ( A e. _V -> ( ( A \ ran g ) X. { z } ) e. _V )
16		vex 3048	. . . . . . . . . 10 |- g e. _V
17	16	cnvex 6740	. . . . . . . . 9 |- `' g e. _V
18	15, 17	jctil 540	. . . . . . . 8 |- ( A e. _V -> ( `' g e. _V /\ ( ( A \ ran g ) X. { z } ) e. _V ) )
19		unexb 6591	. . . . . . . 8 |- ( ( `' g e. _V /\ ( ( A \ ran g ) X. { z } ) e. _V ) <-> ( `' g u. ( ( A \ ran g ) X. { z } ) ) e. _V )
20	18, 19	sylib 200	. . . . . . 7 |- ( A e. _V -> ( `' g u. ( ( A \ ran g ) X. { z } ) ) e. _V )
21		df-f1 5587	. . . . . . . . . . . . 13 |- ( g : B -1-1-> A <-> ( g : B --> A /\ Fun `' g ) )
22	21	simprbi 466	. . . . . . . . . . . 12 |- ( g : B -1-1-> A -> Fun `' g )
23		vex 3048	. . . . . . . . . . . . . 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 541	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ( Fun `' g /\ Fun ( ( A \ ran g ) X. { z } ) ) )
28		df-rn 4845	. . . . . . . . . . . . . 14 |- ran g = dom `' g
29	28	eqcomi 2460	. . . . . . . . . . . . 13 |- dom `' g = ran g
30	23	snnz 4090	. . . . . . . . . . . . . 14 |- { z } =/= (/)
31		dmxp 5053	. . . . . . . . . . . . . 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 3632	. . . . . . . . . . . 12 |- ( dom `' g i^i dom ( ( A \ ran g ) X. { z } ) ) = ( ran g i^i ( A \ ran g ) )
34		disjdif 3839	. . . . . . . . . . . 12 |- ( ran g i^i ( A \ ran g ) ) = (/)
35	33, 34	eqtri 2473	. . . . . . . . . . 11 |- ( dom `' g i^i dom ( ( A \ ran g ) X. { z } ) ) = (/)
36		funun 5624	. . . . . . . . . . 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 668	. . . . . . . . . 10 |- ( g : B -1-1-> A -> Fun ( `' g u. ( ( A \ ran g ) X. { z } ) ) )
38	37	adantl 468	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> Fun ( `' g u. ( ( A \ ran g ) X. { z } ) ) )
39		dmun 5041	. . . . . . . . . . . 12 |- dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( dom `' g u. dom ( ( A \ ran g ) X. { z } ) )
40	28	uneq1i 3584	. . . . . . . . . . . 12 |- ( ran g u. dom ( ( A \ ran g ) X. { z } ) ) = ( dom `' g u. dom ( ( A \ ran g ) X. { z } ) )
41	32	uneq2i 3585	. . . . . . . . . . . 12 |- ( ran g u. dom ( ( A \ ran g ) X. { z } ) ) = ( ran g u. ( A \ ran g ) )
42	39, 40, 41	3eqtr2i 2479	. . . . . . . . . . 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 17	. . . . . . . . . . . 12 |- ( g : B -1-1-> A -> ran g C_ A )
46		undif 3848	. . . . . . . . . . . 12 |- ( ran g C_ A <-> ( ran g u. ( A \ ran g ) ) = A )
47	45, 46	sylib 200	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ( ran g u. ( A \ ran g ) ) = A )
48	42, 47	syl5eq 2497	. . . . . . . . . 10 |- ( g : B -1-1-> A -> dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = A )
49	48	adantl 468	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> dom ( `' g u. ( ( A \ ran g ) X. { z } ) ) = A )
50		df-fn 5585	. . . . . . . . 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 670	. . . . . . . 8 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( `' g u. ( ( A \ ran g ) X. { z } ) ) Fn A )
52		rnun 5244	. . . . . . . . 9 |- ran ( `' g u. ( ( A \ ran g ) X. { z } ) ) = ( ran `' g u. ran ( ( A \ ran g ) X. { z } ) )
53		dfdm4 5027	. . . . . . . . . . . 12 |- dom g = ran `' g
54		f1dm 5783	. . . . . . . . . . . 12 |- ( g : B -1-1-> A -> dom g = B )
55	53, 54	syl5eqr 2499	. . . . . . . . . . 11 |- ( g : B -1-1-> A -> ran `' g = B )
56	55	uneq1d 3587	. . . . . . . . . 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 4848	. . . . . . . . . . . . . . . . 17 |- ( ( A \ ran g ) = (/) -> ( ( A \ ran g ) X. { z } ) = ( (/) X. { z } ) )
58		0xp 4915	. . . . . . . . . . . . . . . . 17 |- ( (/) X. { z } ) = (/)
59	57, 58	syl6eq 2501	. . . . . . . . . . . . . . . 16 |- ( ( A \ ran g ) = (/) -> ( ( A \ ran g ) X. { z } ) = (/) )
60	59	rneqd 5062	. . . . . . . . . . . . . . 15 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) = ran (/) )
61		rn0 5086	. . . . . . . . . . . . . . 15 |- ran (/) = (/)
62	60, 61	syl6eq 2501	. . . . . . . . . . . . . 14 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) = (/) )
63		0ss 3763	. . . . . . . . . . . . . 14 |- (/) C_ B
64	62, 63	syl6eqss 3482	. . . . . . . . . . . . 13 |- ( ( A \ ran g ) = (/) -> ran ( ( A \ ran g ) X. { z } ) C_ B )
65	64	a1d 26	. . . . . . . . . . . 12 |- ( ( A \ ran g ) = (/) -> ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B ) )
66		rnxp 5267	. . . . . . . . . . . . . . 15 |- ( ( A \ ran g ) =/= (/) -> ran ( ( A \ ran g ) X. { z } ) = { z } )
67	66	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> ran ( ( A \ ran g ) X. { z } ) = { z } )
68		snssi 4116	. . . . . . . . . . . . . . 15 |- ( z e. B -> { z } C_ B )
69	68	adantl 468	. . . . . . . . . . . . . 14 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> { z } C_ B )
70	67, 69	eqsstrd 3466	. . . . . . . . . . . . 13 |- ( ( ( A \ ran g ) =/= (/) /\ z e. B ) -> ran ( ( A \ ran g ) X. { z } ) C_ B )
71	70	ex 436	. . . . . . . . . . . 12 |- ( ( A \ ran g ) =/= (/) -> ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B ) )
72	65, 71	pm2.61ine 2707	. . . . . . . . . . 11 |- ( z e. B -> ran ( ( A \ ran g ) X. { z } ) C_ B )
73		ssequn2 3607	. . . . . . . . . . 11 |- ( ran ( ( A \ ran g ) X. { z } ) C_ B <-> ( B u. ran ( ( A \ ran g ) X. { z } ) ) = B )
74	72, 73	sylib 200	. . . . . . . . . 10 |- ( z e. B -> ( B u. ran ( ( A \ ran g ) X. { z } ) ) = B )
75	56, 74	sylan9eqr 2507	. . . . . . . . 9 |- ( ( z e. B /\ g : B -1-1-> A ) -> ( ran `' g u. ran ( ( A \ ran g ) X. { z } ) ) = B )
76	52, 75	syl5eq 2497	. . . . . . . 8 |- ( ( z e. B /\ g : B -1-1-> A ) -> ran ( `' g u. ( ( A \ ran g ) X. { z } ) ) = B )
77		df-fo 5588	. . . . . . . 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 670	. . . . . . 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 3135	. . . . . . 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 39	. . . . . 6 |- ( A e. _V -> ( ( z e. B /\ g : B -1-1-> A ) -> E. f f : A -onto-> B ) )
82	81	expdimp 439	. . . . 5 |- ( ( A e. _V /\ z e. B ) -> ( g : B -1-1-> A -> E. f f : A -onto-> B ) )
83	82	exlimdv 1779	. . . 4 |- ( ( A e. _V /\ z e. B ) -> ( E. g g : B -1-1-> A -> E. f f : A -onto-> B ) )
84	83	ex 436	. . 3 |- ( A e. _V -> ( z e. B -> ( E. g g : B -1-1-> A -> E. f f : A -onto-> B ) ) )
85	84	exlimdv 1779	. 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 63	1 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   E.wex 1663   e. wcel 1887   =/= wne 2622   _Vcvv 3045   \ cdif 3401   u. cun 3402   i^i cin 3403   C_ wss 3404   (/)c0 3731   {csn 3968   class class class wbr 4402   X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835   Fun wfun 5576   Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -onto->wfo 5580   ~<_ cdom 7567   ~< csdm 7568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572

This theorem is referenced by:  pwdom  7724  fodomfib  7851  domwdom  8089  iunfictbso  8545  fodomb  8954  brdom3  8956  konigthlem  8993  1stcfb  20460  ovoliunnul  22460  sigapildsys  28984  carsgclctunlem3  29152  ovoliunnfl  31982  voliunnfl  31984  volsupnfl  31985  nnfoctb  37383  caragenunicl  38345