Theorem lenc 6224

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

Hypothesis

Ref	Expression
lenc.1	⊢ A ∈ V

Assertion

Ref	Expression
lenc	⊢ (M ∈ NC → (M ≤c Nc A ↔ ∃x ∈ M x ⊆ 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 6120	. 2 ⊢ (M ∈ NC ↔ ∃y M = Nc y)
2		ncex 6118	. . . . . . 7 ⊢ Nc y ∈ V
3		ncex 6118	. . . . . . 7 ⊢ Nc A ∈ V
4	2, 3	brlec 6114	. . . . . 6 ⊢ ( Nc y ≤c Nc A ↔ ∃p ∈ Nc y∃q ∈ Nc Ap ⊆ q)
5		elnc 6126	. . . . . . . . . . 11 ⊢ (p ∈ Nc y ↔ p ≈ y)
6		bren 6031	. . . . . . . . . . 11 ⊢ (p ≈ y ↔ ∃f f:p–1-1-onto→y)
7	5, 6	bitri 240	. . . . . . . . . 10 ⊢ (p ∈ Nc y ↔ ∃f f:p–1-1-onto→y)
8		elnc 6126	. . . . . . . . . . 11 ⊢ (q ∈ Nc A ↔ q ≈ A)
9		bren 6031	. . . . . . . . . . 11 ⊢ (q ≈ A ↔ ∃g g:q–1-1-onto→A)
10	8, 9	bitri 240	. . . . . . . . . 10 ⊢ (q ∈ Nc A ↔ ∃g g:q–1-1-onto→A)
11	7, 10	anbi12i 678	. . . . . . . . 9 ⊢ ((p ∈ Nc y ∧ q ∈ Nc A) ↔ (∃f f:p–1-1-onto→y ∧ ∃g g:q–1-1-onto→A))
12		eeanv 1913	. . . . . . . . 9 ⊢ (∃f∃g(f:p–1-1-onto→y ∧ g:q–1-1-onto→A) ↔ (∃f f:p–1-1-onto→y ∧ ∃g g:q–1-1-onto→A))
13	11, 12	bitr4i 243	. . . . . . . 8 ⊢ ((p ∈ Nc y ∧ q ∈ Nc A) ↔ ∃f∃g(f:p–1-1-onto→y ∧ g:q–1-1-onto→A))
14		f1of1 5287	. . . . . . . . . . . . . . . 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 ⊆ q) → g:q–1-1→A)
16		simp3 957	. . . . . . . . . . . . . . 15 ⊢ ((f:p–1-1-onto→y ∧ g:q–1-1-onto→A ∧ p ⊆ q) → p ⊆ q)
17		f1ores 5301	. . . . . . . . . . . . . . 15 ⊢ ((g:q–1-1→A ∧ p ⊆ 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 ⊆ q) → (g ↾ p):p–1-1-onto→(g “ p))
19		f1ocnv 5300	. . . . . . . . . . . . . . 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 ⊆ q) → ◡f:y–1-1-onto→p)
21		f1oco 5309	. . . . . . . . . . . . . 14 ⊢ (((g ↾ p):p–1-1-onto→(g “ p) ∧ ◡f:y–1-1-onto→p) → ((g ↾ p) ∘ ◡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 ⊆ q) → ((g ↾ p) ∘ ◡f):y–1-1-onto→(g “ p))
23		f1ocnv 5300	. . . . . . . . . . . . 13 ⊢ (((g ↾ p) ∘ ◡f):y–1-1-onto→(g “ p) → ◡((g ↾ p) ∘ ◡f):(g “ p)–1-1-onto→y)
24		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ g ∈ V
25		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ p ∈ V
26	24, 25	resex 5118	. . . . . . . . . . . . . . . 16 ⊢ (g ↾ p) ∈ V
27		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ f ∈ V
28	27	cnvex 5103	. . . . . . . . . . . . . . . 16 ⊢ ◡f ∈ V
29	26, 28	coex 4751	. . . . . . . . . . . . . . 15 ⊢ ((g ↾ p) ∘ ◡f) ∈ V
30	29	cnvex 5103	. . . . . . . . . . . . . 14 ⊢ ◡((g ↾ p) ∘ ◡f) ∈ V
31	30	f1oen 6034	. . . . . . . . . . . . 13 ⊢ (◡((g ↾ p) ∘ ◡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 ⊆ q) → (g “ p) ≈ y)
33		elnc 6126	. . . . . . . . . . . 12 ⊢ ((g “ p) ∈ Nc y ↔ (g “ p) ≈ y)
34	32, 33	sylibr 203	. . . . . . . . . . 11 ⊢ ((f:p–1-1-onto→y ∧ g:q–1-1-onto→A ∧ p ⊆ q) → (g “ p) ∈ Nc y)
35		imass2 5025	. . . . . . . . . . . . 13 ⊢ (p ⊆ q → (g “ p) ⊆ (g “ q))
36	35	3ad2ant3 978	. . . . . . . . . . . 12 ⊢ ((f:p–1-1-onto→y ∧ g:q–1-1-onto→A ∧ p ⊆ q) → (g “ p) ⊆ (g “ q))
37		f1ofo 5294	. . . . . . . . . . . . . 14 ⊢ (g:q–1-1-onto→A → g:q–onto→A)
38		foima 5275	. . . . . . . . . . . . . 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 ⊆ q) → (g “ q) = A)
41	36, 40	sseqtrd 3308	. . . . . . . . . . 11 ⊢ ((f:p–1-1-onto→y ∧ g:q–1-1-onto→A ∧ p ⊆ q) → (g “ p) ⊆ A)
42		sseq1 3293	. . . . . . . . . . . 12 ⊢ (x = (g “ p) → (x ⊆ A ↔ (g “ p) ⊆ A))
43	42	rspcev 2956	. . . . . . . . . . 11 ⊢ (((g “ p) ∈ Nc y ∧ (g “ p) ⊆ A) → ∃x ∈ Nc yx ⊆ A)
44	34, 41, 43	syl2anc 642	. . . . . . . . . 10 ⊢ ((f:p–1-1-onto→y ∧ g:q–1-1-onto→A ∧ p ⊆ q) → ∃x ∈ Nc yx ⊆ A)
45	44	3expia 1153	. . . . . . . . 9 ⊢ ((f:p–1-1-onto→y ∧ g:q–1-1-onto→A) → (p ⊆ q → ∃x ∈ Nc yx ⊆ A))
46	45	exlimivv 1635	. . . . . . . 8 ⊢ (∃f∃g(f:p–1-1-onto→y ∧ g:q–1-1-onto→A) → (p ⊆ q → ∃x ∈ Nc yx ⊆ A))
47	13, 46	sylbi 187	. . . . . . 7 ⊢ ((p ∈ Nc y ∧ q ∈ Nc A) → (p ⊆ q → ∃x ∈ Nc yx ⊆ A))
48	47	rexlimivv 2744	. . . . . 6 ⊢ (∃p ∈ Nc y∃q ∈ Nc Ap ⊆ q → ∃x ∈ Nc yx ⊆ A)
49	4, 48	sylbi 187	. . . . 5 ⊢ ( Nc y ≤c Nc A → ∃x ∈ Nc yx ⊆ A)
50		vex 2863	. . . . . . . 8 ⊢ x ∈ V
51		lenc.1	. . . . . . . 8 ⊢ A ∈ V
52	50, 51	nclec 6196	. . . . . . 7 ⊢ (x ⊆ A → Nc x ≤c Nc A)
53	50	eqnc 6128	. . . . . . . . 9 ⊢ ( Nc x = Nc y ↔ x ≈ y)
54		elnc 6126	. . . . . . . . 9 ⊢ (x ∈ Nc y ↔ x ≈ y)
55	53, 54	bitr4i 243	. . . . . . . 8 ⊢ ( Nc x = Nc y ↔ x ∈ Nc y)
56		breq1 4643	. . . . . . . 8 ⊢ ( Nc x = Nc y → ( Nc x ≤c Nc A ↔ Nc y ≤c Nc A))
57	55, 56	sylbir 204	. . . . . . 7 ⊢ (x ∈ Nc y → ( Nc x ≤c Nc A ↔ Nc y ≤c Nc A))
58	52, 57	syl5ib 210	. . . . . 6 ⊢ (x ∈ Nc y → (x ⊆ A → Nc y ≤c Nc A))
59	58	rexlimiv 2733	. . . . 5 ⊢ (∃x ∈ Nc yx ⊆ A → Nc y ≤c Nc A)
60	49, 59	impbii 180	. . . 4 ⊢ ( Nc y ≤c Nc A ↔ ∃x ∈ Nc yx ⊆ A)
61		breq1 4643	. . . . 5 ⊢ (M = Nc y → (M ≤c Nc A ↔ Nc y ≤c Nc A))
62		rexeq 2809	. . . . 5 ⊢ (M = Nc y → (∃x ∈ M x ⊆ A ↔ ∃x ∈ Nc yx ⊆ A))
63	61, 62	bibi12d 312	. . . 4 ⊢ (M = Nc y → ((M ≤c Nc A ↔ ∃x ∈ M x ⊆ A) ↔ ( Nc y ≤c Nc A ↔ ∃x ∈ Nc yx ⊆ A)))
64	60, 63	mpbiri 224	. . 3 ⊢ (M = Nc y → (M ≤c Nc A ↔ ∃x ∈ M x ⊆ A))
65	64	exlimiv 1634	. 2 ⊢ (∃y M = Nc y → (M ≤c Nc A ↔ ∃x ∈ M x ⊆ A))
66	1, 65	sylbi 187	1 ⊢ (M ∈ NC → (M ≤c Nc A ↔ ∃x ∈ M x ⊆ A))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258   class class class wbr 4640   ∘ ccom 4722   “ cima 4723  ^◡ccnv 4772   ↾ cres 4775  –1-1→wf1 4779  –onto→wfo 4780  –1-1-onto→wf1o 4781   ≈ cen 6029   NC cncs 6089   ≤_c clec 6090   Nc cnc 6092

This proof depends on 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-lec 6100  df-nc 6102

This theorem is used by:  ce0lenc1  6240  nchoicelem13  6302