Theorem smobeth 10600

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 9957	. . . . . . 7 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
2		ffun 6709	. . . . . . 7 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun card
4		r1fnon 9781	. . . . . . 7 ⊢ 𝑅1 Fn On
5		fnfun 6638	. . . . . . 7 ⊢ (𝑅1 Fn On → Fun 𝑅1)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ Fun 𝑅1
7		funco 6576	. . . . . 6 ⊢ ((Fun card ∧ Fun 𝑅1) → Fun (card ∘ 𝑅1))
8	3, 6, 7	mp2an 692	. . . . 5 ⊢ Fun (card ∘ 𝑅1)
9		funfn 6566	. . . . 5 ⊢ (Fun (card ∘ 𝑅1) ↔ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1))
10	8, 9	mpbi 230	. . . 4 ⊢ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1)
11		rnco 6241	. . . . 5 ⊢ ran (card ∘ 𝑅1) = ran (card ↾ ran 𝑅1)
12		resss 5988	. . . . . . 7 ⊢ (card ↾ ran 𝑅1) ⊆ card
13	12	rnssi 5920	. . . . . 6 ⊢ ran (card ↾ ran 𝑅1) ⊆ ran card
14		frn 6713	. . . . . . 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 3968	. . . . 5 ⊢ ran (card ↾ ran 𝑅1) ⊆ On
17	11, 16	eqsstri 4005	. . . 4 ⊢ ran (card ∘ 𝑅1) ⊆ On
18		df-f 6535	. . . 4 ⊢ ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ ((card ∘ 𝑅1) Fn dom (card ∘ 𝑅1) ∧ ran (card ∘ 𝑅1) ⊆ On))
19	10, 17, 18	mpbir2an 711	. . 3 ⊢ (card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On
20		dmco 6243	. . . 4 ⊢ dom (card ∘ 𝑅1) = (◡𝑅1 “ dom card)
21	20	feq2i 6698	. . 3 ⊢ ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ (card ∘ 𝑅1):(◡𝑅1 “ dom card)⟶On)
22	19, 21	mpbi 230	. 2 ⊢ (card ∘ 𝑅1):(◡𝑅1 “ dom card)⟶On
23		elpreima 7048	. . . . . . . . 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 6377	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
27	25, 26	sylan 580	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
28	24	simprbi 496	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑅1 “ dom card) → (𝑅1‘𝑥) ∈ dom card)
29	28	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑥) ∈ dom card)
30		r1ord2 9795	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)))
31	30	imp 406	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥))
32	25, 31	sylan 580	. . . . . . 7 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥))
33		ssnum 10053	. . . . . . 7 ⊢ (((𝑅1‘𝑥) ∈ dom card ∧ (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)) → (𝑅1‘𝑦) ∈ dom card)
34	29, 32, 33	syl2anc 584	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ∈ dom card)
35		elpreima 7048	. . . . . . 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 583	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (◡𝑅1 “ dom card))
38	37	rgen2 3184	. . . 4 ⊢ ∀𝑥 ∈ (◡𝑅1 “ dom card)∀𝑦 ∈ 𝑥 𝑦 ∈ (◡𝑅1 “ dom card)
39		dftr5 5233	. . . 4 ⊢ (Tr (◡𝑅1 “ dom card) ↔ ∀𝑥 ∈ (◡𝑅1 “ dom card)∀𝑦 ∈ 𝑥 𝑦 ∈ (◡𝑅1 “ dom card))
40	38, 39	mpbir 231	. . 3 ⊢ Tr (◡𝑅1 “ dom card)
41		cnvimass 6069	. . . . 5 ⊢ (◡𝑅1 “ dom card) ⊆ dom 𝑅1
42		dffn2 6708	. . . . . . 7 ⊢ (𝑅1 Fn On ↔ 𝑅1:On⟶V)
43	4, 42	mpbi 230	. . . . . 6 ⊢ 𝑅1:On⟶V
44	43	fdmi 6717	. . . . 5 ⊢ dom 𝑅1 = On
45	41, 44	sseqtri 4007	. . . 4 ⊢ (◡𝑅1 “ dom card) ⊆ On
46		epweon 7769	. . . 4 ⊢ E We On
47		wess 5640	. . . 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 6355	. . 3 ⊢ (Ord (◡𝑅1 “ dom card) ↔ (Tr (◡𝑅1 “ dom card) ∧ E We (◡𝑅1 “ dom card)))
50	40, 48, 49	mpbir2an 711	. 2 ⊢ Ord (◡𝑅1 “ dom card)
51		r1sdom 9788	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
52	25, 51	sylan 580	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
53		cardsdom2 10002	. . . . . . 7 ⊢ (((𝑅1‘𝑦) ∈ dom card ∧ (𝑅1‘𝑥) ∈ dom card) → ((card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑥)))
54	34, 29, 53	syl2anc 584	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑥)))
55	52, 54	mpbird 257	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)))
56		fvco2 6976	. . . . . 6 ⊢ ((𝑅1 Fn On ∧ 𝑦 ∈ On) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1‘𝑦)))
57	4, 27, 56	sylancr 587	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1‘𝑦)))
58	25	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ On)
59		fvco2 6976	. . . . . 6 ⊢ ((𝑅1 Fn On ∧ 𝑥 ∈ On) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1‘𝑥)))
60	4, 58, 59	sylancr 587	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1‘𝑥)))
61	55, 57, 60	3eltr4d 2849	. . . 4 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥))
62	61	ex 412	. . 3 ⊢ (𝑥 ∈ (◡𝑅1 “ dom card) → (𝑦 ∈ 𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
63	62	adantl 481	. 2 ⊢ ((𝑦 ∈ (◡𝑅1 “ dom card) ∧ 𝑥 ∈ (◡𝑅1 “ dom card)) → (𝑦 ∈ 𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
64	22, 50, 63, 20	issmo 8362	1 ⊢ Smo (card ∘ 𝑅1)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  ∀wral 3051  ∃wrex 3060  Vcvv 3459   ⊆ wss 3926   class class class wbr 5119  Tr wtr 5229   E cep 5552   We wwe 5605  ^◡ccnv 5653  dom cdm 5654  ran crn 5655   ↾ cres 5656   “ cima 5657   ∘ ccom 5658  Ord word 6351  Oncon0 6352  Fun wfun 6525   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  Smo wsmo 8359   ≈ cen 8956   ≺ csdm 8958  𝑅₁cr1 9776  cardccrd 9949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-smo 8360  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-r1 9778  df-card 9953

This theorem is referenced by: (None)