Theorem lenc 6223

Description: Less than or equal condition for the cardinality of a number. (Contributed by SF, 18-Mar-2015.)

Hypothesis

Ref	Expression
lenc.1	|- A e. _V

Assertion

Ref	Expression
lenc	|- (M e. NC -> (M <_c Nc A <-> E.x e. M x C_ A))

Distinct variable groups:   x,M   x,A

Proof of Theorem lenc

Dummy variables f g p q y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elncs 6119	. 2 |- (M e. NC <-> E.y M = Nc y)
2		ncex 6117	. . . . . . 7 |- Nc y e. _V
3		ncex 6117	. . . . . . 7 |- Nc A e. _V
4	2, 3	brlec 6113	. . . . . 6 |- ( Nc y <_c Nc A <-> E.p e. Nc yE.q e. Nc Ap C_ q)
5		elnc 6125	. . . . . . . . . . 11 |- (p e. Nc y <-> p ~~ y)
6		bren 6030	. . . . . . . . . . 11 |- (p ~~ y <-> E.f f:p-1-1-onto->y)
7	5, 6	bitri 240	. . . . . . . . . 10 |- (p e. Nc y <-> E.f f:p-1-1-onto->y)
8		elnc 6125	. . . . . . . . . . 11 |- (q e. Nc A <-> q ~~ A)
9		bren 6030	. . . . . . . . . . 11 |- (q ~~ A <-> E.g g:q-1-1-onto->A)
10	8, 9	bitri 240	. . . . . . . . . 10 |- (q e. Nc A <-> E.g g:q-1-1-onto->A)
11	7, 10	anbi12i 678	. . . . . . . . 9 |- ((p e. Nc y /\ q e. Nc A) <-> (E.f f:p-1-1-onto->y /\ E.g g:q-1-1-onto->A))
12		eeanv 1913	. . . . . . . . 9 |- (E.fE.g(f:p-1-1-onto->y /\ g:q-1-1-onto->A) <-> (E.f f:p-1-1-onto->y /\ E.g g:q-1-1-onto->A))
13	11, 12	bitr4i 243	. . . . . . . 8 |- ((p e. Nc y /\ q e. Nc A) <-> E.fE.g(f:p-1-1-onto->y /\ g:q-1-1-onto->A))
14		f1of1 5286	. . . . . . . . . . . . . . . 16 |- (g:q-1-1-onto->A -> g:q-1-1->A)
15	14	3ad2ant2 977	. . . . . . . . . . . . . . 15 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A /\ p C_ q) -> g:q-1-1->A)
16		simp3 957	. . . . . . . . . . . . . . 15 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A /\ p C_ q) -> p C_ q)
17		f1ores 5300	. . . . . . . . . . . . . . 15 |- ((g:q-1-1->A /\ p C_ q) -> (g |` p):p-1-1-onto->(g"p))
18	15, 16, 17	syl2anc 642	. . . . . . . . . . . . . 14 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A /\ p C_ q) -> (g |` p):p-1-1-onto->(g"p))
19		f1ocnv 5299	. . . . . . . . . . . . . . 15 |- (f:p-1-1-onto->y -> `'f:y-1-1-onto->p)
20	19	3ad2ant1 976	. . . . . . . . . . . . . 14 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A /\ p C_ q) -> `'f:y-1-1-onto->p)
21		f1oco 5308	. . . . . . . . . . . . . 14 |- (((g |` p):p-1-1-onto->(g"p) /\ `'f:y-1-1-onto->p) -> ((g |` p) o. `'f):y-1-1-onto->(g"p))
22	18, 20, 21	syl2anc 642	. . . . . . . . . . . . 13 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A /\ p C_ q) -> ((g |` p) o. `'f):y-1-1-onto->(g"p))
23		f1ocnv 5299	. . . . . . . . . . . . 13 |- (((g |` p) o. `'f):y-1-1-onto->(g"p) -> `'((g |` p) o. `'f):(g"p)-1-1-onto->y)
24		vex 2862	. . . . . . . . . . . . . . . . 17 |- g e. _V
25		vex 2862	. . . . . . . . . . . . . . . . 17 |- p e. _V
26	24, 25	resex 5117	. . . . . . . . . . . . . . . 16 |- (g |` p) e. _V
27		vex 2862	. . . . . . . . . . . . . . . . 17 |- f e. _V
28	27	cnvex 5102	. . . . . . . . . . . . . . . 16 |- `'f e. _V
29	26, 28	coex 4750	. . . . . . . . . . . . . . 15 |- ((g |` p) o. `'f) e. _V
30	29	cnvex 5102	. . . . . . . . . . . . . 14 |- `'((g |` p) o. `'f) e. _V
31	30	f1oen 6033	. . . . . . . . . . . . 13 |- (`'((g |` p) o. `'f):(g"p)-1-1-onto->y -> (g"p) ~~ y)
32	22, 23, 31	3syl 18	. . . . . . . . . . . 12 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A /\ p C_ q) -> (g"p) ~~ y)
33		elnc 6125	. . . . . . . . . . . 12 |- ((g"p) e. Nc y <-> (g"p) ~~ y)
34	32, 33	sylibr 203	. . . . . . . . . . 11 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A /\ p C_ q) -> (g"p) e. Nc y)
35		imass2 5024	. . . . . . . . . . . . 13 |- (p C_ q -> (g"p) C_ (g"q))
36	35	3ad2ant3 978	. . . . . . . . . . . 12 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A /\ p C_ q) -> (g"p) C_ (g"q))
37		f1ofo 5293	. . . . . . . . . . . . . 14 |- (g:q-1-1-onto->A -> g:q-onto->A)
38		foima 5274	. . . . . . . . . . . . . 14 |- (g:q-onto->A -> (g"q) = A)
39	37, 38	syl 15	. . . . . . . . . . . . 13 |- (g:q-1-1-onto->A -> (g"q) = A)
40	39	3ad2ant2 977	. . . . . . . . . . . 12 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A /\ p C_ q) -> (g"q) = A)
41	36, 40	sseqtrd 3307	. . . . . . . . . . 11 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A /\ p C_ q) -> (g"p) C_ A)
42		sseq1 3292	. . . . . . . . . . . 12 |- (x = (g"p) -> (x C_ A <-> (g"p) C_ A))
43	42	rspcev 2955	. . . . . . . . . . 11 |- (((g"p) e. Nc y /\ (g"p) C_ A) -> E.x e. Nc yx C_ A)
44	34, 41, 43	syl2anc 642	. . . . . . . . . 10 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A /\ p C_ q) -> E.x e. Nc yx C_ A)
45	44	3expia 1153	. . . . . . . . 9 |- ((f:p-1-1-onto->y /\ g:q-1-1-onto->A) -> (p C_ q -> E.x e. Nc yx C_ A))
46	45	exlimivv 1635	. . . . . . . 8 |- (E.fE.g(f:p-1-1-onto->y /\ g:q-1-1-onto->A) -> (p C_ q -> E.x e. Nc yx C_ A))
47	13, 46	sylbi 187	. . . . . . 7 |- ((p e. Nc y /\ q e. Nc A) -> (p C_ q -> E.x e. Nc yx C_ A))
48	47	rexlimivv 2743	. . . . . 6 |- (E.p e. Nc yE.q e. Nc Ap C_ q -> E.x e. Nc yx C_ A)
49	4, 48	sylbi 187	. . . . 5 |- ( Nc y <_c Nc A -> E.x e. Nc yx C_ A)
50		vex 2862	. . . . . . . 8 |- x e. _V
51		lenc.1	. . . . . . . 8 |- A e. _V
52	50, 51	nclec 6195	. . . . . . 7 |- (x C_ A -> Nc x <_c Nc A)
53	50	eqnc 6127	. . . . . . . . 9 |- ( Nc x = Nc y <-> x ~~ y)
54		elnc 6125	. . . . . . . . 9 |- (x e. Nc y <-> x ~~ y)
55	53, 54	bitr4i 243	. . . . . . . 8 |- ( Nc x = Nc y <-> x e. Nc y)
56		breq1 4642	. . . . . . . 8 |- ( Nc x = Nc y -> ( Nc x <_c Nc A <-> Nc y <_c Nc A))
57	55, 56	sylbir 204	. . . . . . 7 |- (x e. Nc y -> ( Nc x <_c Nc A <-> Nc y <_c Nc A))
58	52, 57	syl5ib 210	. . . . . 6 |- (x e. Nc y -> (x C_ A -> Nc y <_c Nc A))
59	58	rexlimiv 2732	. . . . 5 |- (E.x e. Nc yx C_ A -> Nc y <_c Nc A)
60	49, 59	impbii 180	. . . 4 |- ( Nc y <_c Nc A <-> E.x e. Nc yx C_ A)
61		breq1 4642	. . . . 5 |- (M = Nc y -> (M <_c Nc A <-> Nc y <_c Nc A))
62		rexeq 2808	. . . . 5 |- (M = Nc y -> (E.x e. M x C_ A <-> E.x e. Nc yx C_ A))
63	61, 62	bibi12d 312	. . . 4 |- (M = Nc y -> ((M <_c Nc A <-> E.x e. M x C_ A) <-> ( Nc y <_c Nc A <-> E.x e. Nc yx C_ A)))
64	60, 63	mpbiri 224	. . 3 |- (M = Nc y -> (M <_c Nc A <-> E.x e. M x C_ A))
65	64	exlimiv 1634	. 2 |- (E.y M = Nc y -> (M <_c Nc A <-> E.x e. M x C_ A))
66	1, 65	sylbi 187	1 |- (M e. NC -> (M <_c Nc A <-> E.x e. M x C_ A))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2615  _Vcvv 2859   C_ wss 3257   class class class wbr 4639   o. ccom 4721  "cima 4722  `'ccnv 4771   |` cres 4774  -1-1->wf1 4778  -onto->wfo 4779  -1-1-onto->wf1o 4780   ~~ cen 6028   NC cncs 6088   <__c clec 6089   Nc cnc 6091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-lec 6099  df-nc 6101

This theorem is referenced by:  ce0lenc1  6239  nchoicelem13  6301