Theorem smobeth 10664

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 10017	. . . . . . 7 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
2		ffun 6710	. . . . . . 7 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun card
4		r1fnon 9766	. . . . . . 7 ⊢ 𝑅1 Fn On
5		fnfun 6637	. . . . . . 7 ⊢ (𝑅1 Fn On → Fun 𝑅1)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ Fun 𝑅1
7		funco 6578	. . . . . 6 ⊢ ((Fun card ∧ Fun 𝑅1) → Fun (card ∘ 𝑅1))
8	3, 6, 7	mp2an 705	. . . . 5 ⊢ Fun (card ∘ 𝑅1)
9		funfn 6568	. . . . 5 ⊢ (Fun (card ∘ 𝑅1) ↔ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1))
10	8, 9	mpbi 233	. . . 4 ⊢ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1)
11		rnco 6252	. . . . 5 ⊢ ran (card ∘ 𝑅1) = ran (card ↾ ran 𝑅1)
12		resss 5992	. . . . . . 7 ⊢ (card ↾ ran 𝑅1) ⊆ card
13	12	rnssi 5922	. . . . . 6 ⊢ ran (card ↾ ran 𝑅1) ⊆ ran card
14		frn 6715	. . . . . . 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 3940	. . . . 5 ⊢ ran (card ↾ ran 𝑅1) ⊆ On
17	11, 16	eqsstri 3977	. . . 4 ⊢ ran (card ∘ 𝑅1) ⊆ On
18		df-f 6541	. . . 4 ⊢ ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ ((card ∘ 𝑅1) Fn dom (card ∘ 𝑅1) ∧ ran (card ∘ 𝑅1) ⊆ On))
19	10, 17, 18	mpbir2an 724	. . 3 ⊢ (card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On
20		dmco 6255	. . . 4 ⊢ dom (card ∘ 𝑅1) = (◡𝑅1 “ dom card)
21	20	feq2i 6699	. . 3 ⊢ ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ (card ∘ 𝑅1):(◡𝑅1 “ dom card)⟶On)
22	19, 21	mpbi 233	. 2 ⊢ (card ∘ 𝑅1):(◡𝑅1 “ dom card)⟶On
23		elpreima 7055	. . . . . . . . 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 502	. . . . . . 7 ⊢ (𝑥 ∈ (◡𝑅1 “ dom card) → 𝑥 ∈ On)
26		onelon 6386	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
27	25, 26	sylan 592	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
28	24	simprbi 503	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑅1 “ dom card) → (𝑅1‘𝑥) ∈ dom card)
29	28	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑥) ∈ dom card)
30		r1ord2 9781	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)))
31	30	imp 412	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥))
32	25, 31	sylan 592	. . . . . . 7 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥))
33		ssnum 10111	. . . . . . 7 ⊢ (((𝑅1‘𝑥) ∈ dom card ∧ (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)) → (𝑅1‘𝑦) ∈ dom card)
34	29, 32, 33	syl2anc 596	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ∈ dom card)
35		elpreima 7055	. . . . . . 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 595	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (◡𝑅1 “ dom card))
38	37	rgen2 3203	. . . 4 ⊢ ∀𝑥 ∈ (◡𝑅1 “ dom card)∀𝑦 ∈ 𝑥 𝑦 ∈ (◡𝑅1 “ dom card)
39		dftr5 5216	. . . 4 ⊢ (Tr (◡𝑅1 “ dom card) ↔ ∀𝑥 ∈ (◡𝑅1 “ dom card)∀𝑦 ∈ 𝑥 𝑦 ∈ (◡𝑅1 “ dom card))
40	38, 39	mpbir 234	. . 3 ⊢ Tr (◡𝑅1 “ dom card)
41		cnvimass 6197	. . . . 5 ⊢ (◡𝑅1 “ dom card) ⊆ dom 𝑅1
42		dffn2 6709	. . . . . . 7 ⊢ (𝑅1 Fn On ↔ 𝑅1:On⟶V)
43	4, 42	mpbi 233	. . . . . 6 ⊢ 𝑅1:On⟶V
44	43	fdmi 6719	. . . . 5 ⊢ dom 𝑅1 = On
45	41, 44	sseqtri 3979	. . . 4 ⊢ (◡𝑅1 “ dom card) ⊆ On
46		epweon 7787	. . . 4 ⊢ E We On
47		wess 5637	. . . 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 6364	. . 3 ⊢ (Ord (◡𝑅1 “ dom card) ↔ (Tr (◡𝑅1 “ dom card) ∧ E We (◡𝑅1 “ dom card)))
50	40, 48, 49	mpbir2an 724	. 2 ⊢ Ord (◡𝑅1 “ dom card)
51		r1sdom 9774	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
52	25, 51	sylan 592	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
53		cardsdom2 10062	. . . . . . 7 ⊢ (((𝑅1‘𝑦) ∈ dom card ∧ (𝑅1‘𝑥) ∈ dom card) → ((card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑥)))
54	34, 29, 53	syl2anc 596	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑥)))
55	52, 54	mpbird 260	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)))
56		fvco2 6980	. . . . . 6 ⊢ ((𝑅1 Fn On ∧ 𝑦 ∈ On) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1‘𝑦)))
57	4, 27, 56	sylancr 599	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1‘𝑦)))
58	25	adantr 486	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ On)
59		fvco2 6980	. . . . . 6 ⊢ ((𝑅1 Fn On ∧ 𝑥 ∈ On) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1‘𝑥)))
60	4, 58, 59	sylancr 599	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1‘𝑥)))
61	55, 57, 60	3eltr4d 2876	. . . 4 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥))
62	61	ex 418	. . 3 ⊢ (𝑥 ∈ (◡𝑅1 “ dom card) → (𝑦 ∈ 𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
63	62	adantl 487	. 2 ⊢ ((𝑦 ∈ (◡𝑅1 “ dom card) ∧ 𝑥 ∈ (◡𝑅1 “ dom card)) → (𝑦 ∈ 𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
64	22, 50, 63, 20	issmo 8349	1 ⊢ Smo (card ∘ 𝑅1)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ⊆ wss 3899   class class class wbr 5103  Tr wtr 5212   E cep 5550   We wwe 5603  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  Ord word 6360  Oncon0 6361  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  Smo wsmo 8346   ≈ cen 8963   ≺ csdm 8965  𝑅₁cr1 9759  cardccrd 10009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-smo 8347  df-recs 8372  df-rdg 8411  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-r1 9761  df-card 10013

This theorem is used by: (None)