Theorem fodomr 5547

Description: There exists a mapping from a set onto any (non-empty) set that it dominates.

Assertion

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

Distinct variable groups:   A,f   B,f

Proof of Theorem fodomr

Step	Hyp	Ref	Expression
1		elisset 2299	. . 3 |- (A e. C -> A e. _V)
2	1	3ad2ant1 897	. 2 |- ((A e. C /\ (/) ~< B /\ B ~<_ A) -> A e. _V)
3		reldom 5432	. . . . . 6 |- Rel ~<_
4	3	brrelexi 4029	. . . . 5 |- (B ~<_ A -> B e. _V)
5		0sdomg 5529	. . . . . 6 |- (B e. _V -> ((/) ~< B <-> B =/= (/)))
6		n0 2884	. . . . . 6 |- (B =/= (/) <-> E.z z e. B)
7	5, 6	syl6bb 595	. . . . 5 |- (B e. _V -> ((/) ~< B <-> E.z z e. B))
8	4, 7	syl 12	. . . 4 |- (B ~<_ A -> ((/) ~< B <-> E.z z e. B))
9	8	biimpac 462	. . 3 |- (((/) ~< B /\ B ~<_ A) -> E.z z e. B)
10	9	3adant1 894	. 2 |- ((A e. C /\ (/) ~< B /\ B ~<_ A) -> E.z z e. B)
11		brdomg 5435	. . . 4 |- (A e. C -> (B ~<_ A <-> E.g g:B-1-1->A))
12	11	biimpa 460	. . 3 |- ((A e. C /\ B ~<_ A) -> E.g g:B-1-1->A)
13	12	3adant2 895	. 2 |- ((A e. C /\ (/) ~< B /\ B ~<_ A) -> E.g g:B-1-1->A)
14		xpexg 4095	. . . . . . . . . . 11 |- (((A \ ran g) e. _V /\ {z} e. _V) -> ((A \ ran g) X. {z}) e. _V)
15		difexg 3458	. . . . . . . . . . 11 |- (A e. _V -> (A \ ran g) e. _V)
16		snex 3492	. . . . . . . . . . 11 |- {z} e. _V
17	14, 15, 16	sylancl 525	. . . . . . . . . 10 |- (A e. _V -> ((A \ ran g) X. {z}) e. _V)
18		visset 2295	. . . . . . . . . . 11 |- g e. _V
19	18	cnvex 4425	. . . . . . . . . 10 |- `'g e. _V
20	17, 19	jctil 316	. . . . . . . . 9 |- (A e. _V -> (`'g e. _V /\ ((A \ ran g) X. {z}) e. _V))
21		unexb 3797	. . . . . . . . 9 |- ((`'g e. _V /\ ((A \ ran g) X. {z}) e. _V) <-> (`'g u. ((A \ ran g) X. {z})) e. _V)
22	20, 21	sylib 215	. . . . . . . 8 |- (A e. _V -> (`'g u. ((A \ ran g) X. {z})) e. _V)
23		foeq1 4613	. . . . . . . . 9 |- (f = (`'g u. ((A \ ran g) X. {z})) -> (f:A-onto->B <-> (`'g u. ((A \ ran g) X. {z})):A-onto->B))
24	23	cla4egv 2365	. . . . . . . 8 |- ((`'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))
25	22, 24	syl 12	. . . . . . 7 |- (A e. _V -> ((`'g u. ((A \ ran g) X. {z})):A-onto->B -> E.f f:A-onto->B))
26		df-fo 4012	. . . . . . . 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))
27		df-fn 4009	. . . . . . . . 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))
28		funun 4462	. . . . . . . . . . 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})))
29		df-f1 4011	. . . . . . . . . . . . 13 |- (g:B-1-1->A <-> (g:B-->A /\ Fun `'g))
30	29	simprbi 353	. . . . . . . . . . . 12 |- (g:B-1-1->A -> Fun `'g)
31		visset 2295	. . . . . . . . . . . . . 14 |- z e. _V
32	31	fconst 4602	. . . . . . . . . . . . 13 |- ((A \ ran g) X. {z}):(A \ ran g)-->{z}
33		ffun 4565	. . . . . . . . . . . . 13 |- (((A \ ran g) X. {z}):(A \ ran g)-->{z} -> Fun ((A \ ran g) X. {z}))
34	32, 33	ax-mp 7	. . . . . . . . . . . 12 |- Fun ((A \ ran g) X. {z})
35	30, 34	jctir 317	. . . . . . . . . . 11 |- (g:B-1-1->A -> (Fun `'g /\ Fun ((A \ ran g) X. {z})))
36		df-rn 4005	. . . . . . . . . . . . . 14 |- ran g = dom `' g
37	36	eqcomi 1888	. . . . . . . . . . . . 13 |- dom `' g = ran g
38	31	snnz 3119	. . . . . . . . . . . . . 14 |- {z} =/= (/)
39		dmxp 4177	. . . . . . . . . . . . . 14 |- ({z} =/= (/) -> dom ((A \ ran g) X. {z}) = (A \ ran g))
40	38, 39	ax-mp 7	. . . . . . . . . . . . 13 |- dom ((A \ ran g) X. {z}) = (A \ ran g)
41	37, 40	ineq12i 2794	. . . . . . . . . . . 12 |- (dom `' g i^i dom ((A \ ran g) X. {z})) = (ran g i^i (A \ ran g))
42		difdisj 2945	. . . . . . . . . . . 12 |- (ran g i^i (A \ ran g)) = (/)
43	41, 42	eqtri 1908	. . . . . . . . . . 11 |- (dom `' g i^i dom ((A \ ran g) X. {z})) = (/)
44	28, 35, 43	sylancl 525	. . . . . . . . . 10 |- (g:B-1-1->A -> Fun (`'g u. ((A \ ran g) X. {z})))
45	44	adantl 424	. . . . . . . . 9 |- ((z e. B /\ g:B-1-1->A) -> Fun (`'g u. ((A \ ran g) X. {z})))
46		f1f 4610	. . . . . . . . . . . . 13 |- (g:B-1-1->A -> g:B-->A)
47		frn 4569	. . . . . . . . . . . . 13 |- (g:B-->A -> ran g C_ A)
48	46, 47	syl 12	. . . . . . . . . . . 12 |- (g:B-1-1->A -> ran g C_ A)
49		undif 2954	. . . . . . . . . . . 12 |- (ran g C_ A <-> (ran g u. (A \ ran g)) = A)
50	48, 49	sylib 215	. . . . . . . . . . 11 |- (g:B-1-1->A -> (ran g u. (A \ ran g)) = A)
51		dmun 4163	. . . . . . . . . . . 12 |- dom (`'g u. ((A \ ran g) X. {z})) = (dom `' g u. dom ((A \ ran g) X. {z}))
52	36	uneq1i 2751	. . . . . . . . . . . 12 |- (ran g u. dom ((A \ ran g) X. {z})) = (dom `' g u. dom ((A \ ran g) X. {z}))
53	40	uneq2i 2752	. . . . . . . . . . . 12 |- (ran g u. dom ((A \ ran g) X. {z})) = (ran g u. (A \ ran g))
54	51, 52, 53	3eqtr2i 1915	. . . . . . . . . . 11 |- dom (`'g u. ((A \ ran g) X. {z})) = (ran g u. (A \ ran g))
55	50, 54	syl5eq 1940	. . . . . . . . . 10 |- (g:B-1-1->A -> dom (`'g u. ((A \ ran g) X. {z})) = A)
56	55	adantl 424	. . . . . . . . 9 |- ((z e. B /\ g:B-1-1->A) -> dom (`'g u. ((A \ ran g) X. {z})) = A)
57	27, 45, 56	sylanbrc 527	. . . . . . . 8 |- ((z e. B /\ g:B-1-1->A) -> (`'g u. ((A \ ran g) X. {z})) Fn A)
58		fdm 4567	. . . . . . . . . . . . 13 |- (g:B-->A -> dom g = B)
59	46, 58	syl 12	. . . . . . . . . . . 12 |- (g:B-1-1->A -> dom g = B)
60		dfdm4 4151	. . . . . . . . . . . 12 |- dom g = ran `' g
61	59, 60	syl5eqr 1942	. . . . . . . . . . 11 |- (g:B-1-1->A -> ran `' g = B)
62	61	uneq1d 2754	. . . . . . . . . 10 |- (g:B-1-1->A -> (ran `' g u. ran ((A \ ran g) X. {z})) = (B u. ran ((A \ ran g) X. {z})))
63		0ss 2900	. . . . . . . . . . . . . 14 |- (/) C_ B
64		xpeq1 4016	. . . . . . . . . . . . . . . . . 18 |- ((A \ ran g) = (/) -> ((A \ ran g) X. {z}) = ((/) X. {z}))
65		xp0r 4065	. . . . . . . . . . . . . . . . . 18 |- ((/) X. {z}) = (/)
66	64, 65	syl6eq 1944	. . . . . . . . . . . . . . . . 17 |- ((A \ ran g) = (/) -> ((A \ ran g) X. {z}) = (/))
67	66	rneqd 4188	. . . . . . . . . . . . . . . 16 |- ((A \ ran g) = (/) -> ran ((A \ ran g) X. {z}) = ran (/))
68		rn0 4203	. . . . . . . . . . . . . . . 16 |- ran (/) = (/)
69	67, 68	syl6eq 1944	. . . . . . . . . . . . . . 15 |- ((A \ ran g) = (/) -> ran ((A \ ran g) X. {z}) = (/))
70	69	sseq1d 2644	. . . . . . . . . . . . . 14 |- ((A \ ran g) = (/) -> (ran ((A \ ran g) X. {z}) C_ B <-> (/) C_ B))
71	63, 70	mpbiri 211	. . . . . . . . . . . . 13 |- ((A \ ran g) = (/) -> ran ((A \ ran g) X. {z}) C_ B)
72	71	a1d 15	. . . . . . . . . . . 12 |- ((A \ ran g) = (/) -> (z e. B -> ran ((A \ ran g) X. {z}) C_ B))
73		rnxp 4342	. . . . . . . . . . . . . . 15 |- ((A \ ran g) =/= (/) -> ran ((A \ ran g) X. {z}) = {z})
74	73	adantr 425	. . . . . . . . . . . . . 14 |- (((A \ ran g) =/= (/) /\ z e. B) -> ran ((A \ ran g) X. {z}) = {z})
75		snssi 3129	. . . . . . . . . . . . . . 15 |- (z e. B -> {z} C_ B)
76	75	adantl 424	. . . . . . . . . . . . . 14 |- (((A \ ran g) =/= (/) /\ z e. B) -> {z} C_ B)
77	74, 76	eqsstrd 2651	. . . . . . . . . . . . 13 |- (((A \ ran g) =/= (/) /\ z e. B) -> ran ((A \ ran g) X. {z}) C_ B)
78	77	ex 402	. . . . . . . . . . . 12 |- ((A \ ran g) =/= (/) -> (z e. B -> ran ((A \ ran g) X. {z}) C_ B))
79	72, 78	pm2.61ine 2089	. . . . . . . . . . 11 |- (z e. B -> ran ((A \ ran g) X. {z}) C_ B)
80		ssequn2 2779	. . . . . . . . . . 11 |- (ran ((A \ ran g) X. {z}) C_ B <-> (B u. ran ((A \ ran g) X. {z})) = B)
81	79, 80	sylib 215	. . . . . . . . . 10 |- (z e. B -> (B u. ran ((A \ ran g) X. {z})) = B)
82	62, 81	sylan9eqr 1951	. . . . . . . . 9 |- ((z e. B /\ g:B-1-1->A) -> (ran `' g u. ran ((A \ ran g) X. {z})) = B)
83		rnun 4325	. . . . . . . . 9 |- ran (`'g u. ((A \ ran g) X. {z})) = (ran `' g u. ran ((A \ ran g) X. {z}))
84	82, 83	syl5eq 1940	. . . . . . . 8 |- ((z e. B /\ g:B-1-1->A) -> ran (`'g u. ((A \ ran g) X. {z})) = B)
85	26, 57, 84	sylanbrc 527	. . . . . . 7 |- ((z e. B /\ g:B-1-1->A) -> (`'g u. ((A \ ran g) X. {z})):A-onto->B)
86	25, 85	syl5 20	. . . . . 6 |- (A e. _V -> ((z e. B /\ g:B-1-1->A) -> E.f f:A-onto->B))
87	86	expdimp 406	. . . . 5 |- ((A e. _V /\ z e. B) -> (g:B-1-1->A -> E.f f:A-onto->B))
88	87	19.23adv 1584	. . . 4 |- ((A e. _V /\ z e. B) -> (E.g g:B-1-1->A -> E.f f:A-onto->B))
89	88	ex 402	. . 3 |- (A e. _V -> (z e. B -> (E.g g:B-1-1->A -> E.f f:A-onto->B)))
90	89	19.23adv 1584	. 2 |- (A e. _V -> (E.z z e. B -> (E.g g:B-1-1->A -> E.f f:A-onto->B)))
91	2, 10, 13, 90	syl3c 84	1 |- ((A e. C /\ (/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  _Vcvv 2292   \ cdif 2590   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044   class class class wbr 3338   X. cxp 3984  `'ccnv 3985  dom cdm 3986  ran crn 3987  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  -onto->wfo 3996   ~<_ cdom 5424   ~< csdm 5425

This theorem is referenced by:  fodomfib 5657  fodomb 5962  brdom3 5963

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429

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