Theorem smobeth 10507

Description: The beth function is strictly monotone. This function is not strictly the beth function, but rather beth_A is the same as (card‘(𝑅₁‘(ω +_o 𝐴))), since conventionally we start counting at the first infinite level, and ignore the finite levels. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 2-Jun-2015.)

Assertion

Ref	Expression
smobeth	⊢ Smo (card ∘ 𝑅1)

Proof of Theorem smobeth

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardf2 9865	. . . . . . 7 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
2		ffun 6665	. . . . . . 7 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun card
4		r1fnon 9689	. . . . . . 7 ⊢ 𝑅1 Fn On
5		fnfun 6592	. . . . . . 7 ⊢ (𝑅1 Fn On → Fun 𝑅1)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ Fun 𝑅1
7		funco 6532	. . . . . 6 ⊢ ((Fun card ∧ Fun 𝑅1) → Fun (card ∘ 𝑅1))
8	3, 6, 7	mp2an 698	. . . . 5 ⊢ Fun (card ∘ 𝑅1)
9		funfn 6522	. . . . 5 ⊢ (Fun (card ∘ 𝑅1) ↔ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1))
10	8, 9	mpbi 231	. . . 4 ⊢ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1)
11		rnco 6210	. . . . 5 ⊢ ran (card ∘ 𝑅1) = ran (card ↾ ran 𝑅1)
12		resss 5960	. . . . . . 7 ⊢ (card ↾ ran 𝑅1) ⊆ card
13	12	rnssi 5889	. . . . . 6 ⊢ ran (card ↾ ran 𝑅1) ⊆ ran card
14		frn 6669	. . . . . . 7 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → ran card ⊆ On)
15	1, 14	ax-mp 5	. . . . . 6 ⊢ ran card ⊆ On
16	13, 15	sstri 3931	. . . . 5 ⊢ ran (card ↾ ran 𝑅1) ⊆ On
17	11, 16	eqsstri 3968	. . . 4 ⊢ ran (card ∘ 𝑅1) ⊆ On
18		df-f 6496	. . . 4 ⊢ ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ ((card ∘ 𝑅1) Fn dom (card ∘ 𝑅1) ∧ ran (card ∘ 𝑅1) ⊆ On))
19	10, 17, 18	mpbir2an 717	. . 3 ⊢ (card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On
20		dmco 6213	. . . 4 ⊢ dom (card ∘ 𝑅1) = (◡𝑅1 “ dom card)
21	20	feq2i 6654	. . 3 ⊢ ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ (card ∘ 𝑅1):(◡𝑅1 “ dom card)⟶On)
22	19, 21	mpbi 231	. 2 ⊢ (card ∘ 𝑅1):(◡𝑅1 “ dom card)⟶On
23		elpreima 7006	. . . . . . . . 9 ⊢ (𝑅1 Fn On → (𝑥 ∈ (◡𝑅1 “ dom card) ↔ (𝑥 ∈ On ∧ (𝑅1‘𝑥) ∈ dom card)))
24	4, 23	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑅1 “ dom card) ↔ (𝑥 ∈ On ∧ (𝑅1‘𝑥) ∈ dom card))
25	24	simplbi 497	. . . . . . 7 ⊢ (𝑥 ∈ (◡𝑅1 “ dom card) → 𝑥 ∈ On)
26		onelon 6342	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
27	25, 26	sylan 586	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
28	24	simprbi 498	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑅1 “ dom card) → (𝑅1‘𝑥) ∈ dom card)
29	28	adantr 481	. . . . . . 7 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑥) ∈ dom card)
30		r1ord2 9703	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)))
31	30	imp 407	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥))
32	25, 31	sylan 586	. . . . . . 7 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥))
33		ssnum 9959	. . . . . . 7 ⊢ (((𝑅1‘𝑥) ∈ dom card ∧ (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)) → (𝑅1‘𝑦) ∈ dom card)
34	29, 32, 33	syl2anc 590	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ∈ dom card)
35		elpreima 7006	. . . . . . 7 ⊢ (𝑅1 Fn On → (𝑦 ∈ (◡𝑅1 “ dom card) ↔ (𝑦 ∈ On ∧ (𝑅1‘𝑦) ∈ dom card)))
36	4, 35	ax-mp 5	. . . . . 6 ⊢ (𝑦 ∈ (◡𝑅1 “ dom card) ↔ (𝑦 ∈ On ∧ (𝑅1‘𝑦) ∈ dom card))
37	27, 34, 36	sylanbrc 589	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (◡𝑅1 “ dom card))
38	37	rgen2 3180	. . . 4 ⊢ ∀𝑥 ∈ (◡𝑅1 “ dom card)∀𝑦 ∈ 𝑥 𝑦 ∈ (◡𝑅1 “ dom card)
39		dftr5 5190	. . . 4 ⊢ (Tr (◡𝑅1 “ dom card) ↔ ∀𝑥 ∈ (◡𝑅1 “ dom card)∀𝑦 ∈ 𝑥 𝑦 ∈ (◡𝑅1 “ dom card))
40	38, 39	mpbir 232	. . 3 ⊢ Tr (◡𝑅1 “ dom card)
41		cnvimass 6041	. . . . 5 ⊢ (◡𝑅1 “ dom card) ⊆ dom 𝑅1
42		dffn2 6664	. . . . . . 7 ⊢ (𝑅1 Fn On ↔ 𝑅1:On⟶V)
43	4, 42	mpbi 231	. . . . . 6 ⊢ 𝑅1:On⟶V
44	43	fdmi 6673	. . . . 5 ⊢ dom 𝑅1 = On
45	41, 44	sseqtri 3970	. . . 4 ⊢ (◡𝑅1 “ dom card) ⊆ On
46		epweon 7725	. . . 4 ⊢ E We On
47		wess 5611	. . . 4 ⊢ ((◡𝑅1 “ dom card) ⊆ On → ( E We On → E We (◡𝑅1 “ dom card)))
48	45, 46, 47	mp2 9	. . 3 ⊢ E We (◡𝑅1 “ dom card)
49		df-ord 6320	. . 3 ⊢ (Ord (◡𝑅1 “ dom card) ↔ (Tr (◡𝑅1 “ dom card) ∧ E We (◡𝑅1 “ dom card)))
50	40, 48, 49	mpbir2an 717	. 2 ⊢ Ord (◡𝑅1 “ dom card)
51		r1sdom 9696	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
52	25, 51	sylan 586	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
53		cardsdom2 9910	. . . . . . 7 ⊢ (((𝑅1‘𝑦) ∈ dom card ∧ (𝑅1‘𝑥) ∈ dom card) → ((card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑥)))
54	34, 29, 53	syl2anc 590	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑥)))
55	52, 54	mpbird 258	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)))
56		fvco2 6931	. . . . . 6 ⊢ ((𝑅1 Fn On ∧ 𝑦 ∈ On) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1‘𝑦)))
57	4, 27, 56	sylancr 593	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1‘𝑦)))
58	25	adantr 481	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ On)
59		fvco2 6931	. . . . . 6 ⊢ ((𝑅1 Fn On ∧ 𝑥 ∈ On) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1‘𝑥)))
60	4, 58, 59	sylancr 593	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1‘𝑥)))
61	55, 57, 60	3eltr4d 2855	. . . 4 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥))
62	61	ex 413	. . 3 ⊢ (𝑥 ∈ (◡𝑅1 “ dom card) → (𝑦 ∈ 𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
63	62	adantl 482	. 2 ⊢ ((𝑦 ∈ (◡𝑅1 “ dom card) ∧ 𝑥 ∈ (◡𝑅1 “ dom card)) → (𝑦 ∈ 𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
64	22, 50, 63, 20	issmo 8285	1 ⊢ Smo (card ∘ 𝑅1)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2718  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ⊆ wss 3890   class class class wbr 5079  Tr wtr 5186   E cep 5524   We wwe 5577  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628   ∘ ccom 5629  Ord word 6316  Oncon0 6317  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  Smo wsmo 8282   ≈ cen 8887   ≺ csdm 8889  𝑅₁cr1 9684  cardccrd 9857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-smo 8283  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-r1 9686  df-card 9861

This theorem is referenced by: (None)