Theorem nclenc 6223

Description: Comparison rule for cardinalities. (Contributed by SF, 24-Mar-2015.)

Hypotheses

Ref	Expression
nclenc.1	⊢ A ∈ V
nclenc.2	⊢ B ∈ V

Assertion

Ref	Expression
nclenc	⊢ ( Nc A ≤c Nc B ↔ ∃f f:A–1-1→B)

Distinct variable groups:   A,f   B,f

Proof of Theorem nclenc

Dummy variables a b g h i p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nclenc.1	. . . . 5 ⊢ A ∈ V
2	1	ncelncsi 6122	. . . 4 ⊢ Nc A ∈ NC
3		nclenc.2	. . . . 5 ⊢ B ∈ V
4	3	ncelncsi 6122	. . . 4 ⊢ Nc B ∈ NC
5		dflec3 6222	. . . 4 ⊢ (( Nc A ∈ NC ∧ Nc B ∈ NC ) → ( Nc A ≤c Nc B ↔ ∃p ∈ Nc A∃q ∈ Nc B∃g g:p–1-1→q))
6	2, 4, 5	mp2an 653	. . 3 ⊢ ( Nc A ≤c Nc B ↔ ∃p ∈ Nc A∃q ∈ Nc B∃g g:p–1-1→q)
7		elnc 6126	. . . . . . . . 9 ⊢ (p ∈ Nc A ↔ p ≈ A)
8		bren 6031	. . . . . . . . 9 ⊢ (p ≈ A ↔ ∃h h:p–1-1-onto→A)
9	7, 8	bitri 240	. . . . . . . 8 ⊢ (p ∈ Nc A ↔ ∃h h:p–1-1-onto→A)
10		elnc 6126	. . . . . . . . 9 ⊢ (q ∈ Nc B ↔ q ≈ B)
11		bren 6031	. . . . . . . . 9 ⊢ (q ≈ B ↔ ∃i i:q–1-1-onto→B)
12	10, 11	bitri 240	. . . . . . . 8 ⊢ (q ∈ Nc B ↔ ∃i i:q–1-1-onto→B)
13	9, 12	anbi12i 678	. . . . . . 7 ⊢ ((p ∈ Nc A ∧ q ∈ Nc B) ↔ (∃h h:p–1-1-onto→A ∧ ∃i i:q–1-1-onto→B))
14		eeanv 1913	. . . . . . 7 ⊢ (∃h∃i(h:p–1-1-onto→A ∧ i:q–1-1-onto→B) ↔ (∃h h:p–1-1-onto→A ∧ ∃i i:q–1-1-onto→B))
15	13, 14	bitr4i 243	. . . . . 6 ⊢ ((p ∈ Nc A ∧ q ∈ Nc B) ↔ ∃h∃i(h:p–1-1-onto→A ∧ i:q–1-1-onto→B))
16		f1of1 5287	. . . . . . . . . . . 12 ⊢ (i:q–1-1-onto→B → i:q–1-1→B)
17	16	3ad2ant2 977	. . . . . . . . . . 11 ⊢ ((h:p–1-1-onto→A ∧ i:q–1-1-onto→B ∧ g:p–1-1→q) → i:q–1-1→B)
18		simp3 957	. . . . . . . . . . 11 ⊢ ((h:p–1-1-onto→A ∧ i:q–1-1-onto→B ∧ g:p–1-1→q) → g:p–1-1→q)
19		f1co 5265	. . . . . . . . . . 11 ⊢ ((i:q–1-1→B ∧ g:p–1-1→q) → (i ∘ g):p–1-1→B)
20	17, 18, 19	syl2anc 642	. . . . . . . . . 10 ⊢ ((h:p–1-1-onto→A ∧ i:q–1-1-onto→B ∧ g:p–1-1→q) → (i ∘ g):p–1-1→B)
21		f1ocnv 5300	. . . . . . . . . . . 12 ⊢ (h:p–1-1-onto→A → ◡h:A–1-1-onto→p)
22		f1of1 5287	. . . . . . . . . . . 12 ⊢ (◡h:A–1-1-onto→p → ◡h:A–1-1→p)
23	21, 22	syl 15	. . . . . . . . . . 11 ⊢ (h:p–1-1-onto→A → ◡h:A–1-1→p)
24	23	3ad2ant1 976	. . . . . . . . . 10 ⊢ ((h:p–1-1-onto→A ∧ i:q–1-1-onto→B ∧ g:p–1-1→q) → ◡h:A–1-1→p)
25		f1co 5265	. . . . . . . . . 10 ⊢ (((i ∘ g):p–1-1→B ∧ ◡h:A–1-1→p) → ((i ∘ g) ∘ ◡h):A–1-1→B)
26	20, 24, 25	syl2anc 642	. . . . . . . . 9 ⊢ ((h:p–1-1-onto→A ∧ i:q–1-1-onto→B ∧ g:p–1-1→q) → ((i ∘ g) ∘ ◡h):A–1-1→B)
27		vex 2863	. . . . . . . . . . . 12 ⊢ i ∈ V
28		vex 2863	. . . . . . . . . . . 12 ⊢ g ∈ V
29	27, 28	coex 4751	. . . . . . . . . . 11 ⊢ (i ∘ g) ∈ V
30		vex 2863	. . . . . . . . . . . 12 ⊢ h ∈ V
31	30	cnvex 5103	. . . . . . . . . . 11 ⊢ ◡h ∈ V
32	29, 31	coex 4751	. . . . . . . . . 10 ⊢ ((i ∘ g) ∘ ◡h) ∈ V
33		f1eq1 5254	. . . . . . . . . 10 ⊢ (f = ((i ∘ g) ∘ ◡h) → (f:A–1-1→B ↔ ((i ∘ g) ∘ ◡h):A–1-1→B))
34	32, 33	spcev 2947	. . . . . . . . 9 ⊢ (((i ∘ g) ∘ ◡h):A–1-1→B → ∃f f:A–1-1→B)
35	26, 34	syl 15	. . . . . . . 8 ⊢ ((h:p–1-1-onto→A ∧ i:q–1-1-onto→B ∧ g:p–1-1→q) → ∃f f:A–1-1→B)
36	35	3expia 1153	. . . . . . 7 ⊢ ((h:p–1-1-onto→A ∧ i:q–1-1-onto→B) → (g:p–1-1→q → ∃f f:A–1-1→B))
37	36	exlimivv 1635	. . . . . 6 ⊢ (∃h∃i(h:p–1-1-onto→A ∧ i:q–1-1-onto→B) → (g:p–1-1→q → ∃f f:A–1-1→B))
38	15, 37	sylbi 187	. . . . 5 ⊢ ((p ∈ Nc A ∧ q ∈ Nc B) → (g:p–1-1→q → ∃f f:A–1-1→B))
39	38	exlimdv 1636	. . . 4 ⊢ ((p ∈ Nc A ∧ q ∈ Nc B) → (∃g g:p–1-1→q → ∃f f:A–1-1→B))
40	39	rexlimivv 2744	. . 3 ⊢ (∃p ∈ Nc A∃q ∈ Nc B∃g g:p–1-1→q → ∃f f:A–1-1→B)
41	6, 40	sylbi 187	. 2 ⊢ ( Nc A ≤c Nc B → ∃f f:A–1-1→B)
42	1	ncid 6124	. . . 4 ⊢ A ∈ Nc A
43	3	ncid 6124	. . . 4 ⊢ B ∈ Nc B
44		f1eq2 5255	. . . . . 6 ⊢ (a = A → (f:a–1-1→b ↔ f:A–1-1→b))
45	44	exbidv 1626	. . . . 5 ⊢ (a = A → (∃f f:a–1-1→b ↔ ∃f f:A–1-1→b))
46		f1eq3 5256	. . . . . 6 ⊢ (b = B → (f:A–1-1→b ↔ f:A–1-1→B))
47	46	exbidv 1626	. . . . 5 ⊢ (b = B → (∃f f:A–1-1→b ↔ ∃f f:A–1-1→B))
48	45, 47	rspc2ev 2964	. . . 4 ⊢ ((A ∈ Nc A ∧ B ∈ Nc B ∧ ∃f f:A–1-1→B) → ∃a ∈ Nc A∃b ∈ Nc B∃f f:a–1-1→b)
49	42, 43, 48	mp3an12 1267	. . 3 ⊢ (∃f f:A–1-1→B → ∃a ∈ Nc A∃b ∈ Nc B∃f f:a–1-1→b)
50		dflec3 6222	. . . 4 ⊢ (( Nc A ∈ NC ∧ Nc B ∈ NC ) → ( Nc A ≤c Nc B ↔ ∃a ∈ Nc A∃b ∈ Nc B∃f f:a–1-1→b))
51	2, 4, 50	mp2an 653	. . 3 ⊢ ( Nc A ≤c Nc B ↔ ∃a ∈ Nc A∃b ∈ Nc B∃f f:a–1-1→b)
52	49, 51	sylibr 203	. 2 ⊢ (∃f f:A–1-1→B → Nc A ≤c Nc B)
53	41, 52	impbii 180	1 ⊢ ( Nc A ≤c Nc B ↔ ∃f f:A–1-1→B)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   class class class wbr 4640   ∘ ccom 4722  ^◡ccnv 4772  –1-1→wf1 4779  –1-1-onto→wf1o 4781   ≈ cen 6029   NC cncs 6089   ≤_c clec 6090   Nc cnc 6092

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 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 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 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 referenced by: (None)