Theorem smobeth 10582

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 9941	. . . . . . 7 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
2		ffun 6712	. . . . . . 7 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun card
4		r1fnon 9742	. . . . . . 7 ⊢ 𝑅1 Fn On
5		fnfun 6639	. . . . . . 7 ⊢ (𝑅1 Fn On → Fun 𝑅1)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ Fun 𝑅1
7		funco 6580	. . . . . 6 ⊢ ((Fun card ∧ Fun 𝑅1) → Fun (card ∘ 𝑅1))
8	3, 6, 7	mp2an 705	. . . . 5 ⊢ Fun (card ∘ 𝑅1)
9		funfn 6570	. . . . 5 ⊢ (Fun (card ∘ 𝑅1) ↔ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1))
10	8, 9	mpbi 233	. . . 4 ⊢ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1)
11		rnco 6255	. . . . 5 ⊢ ran (card ∘ 𝑅1) = ran (card ↾ ran 𝑅1)
12		resss 6002	. . . . . . 7 ⊢ (card ↾ ran 𝑅1) ⊆ card
13	12	rnssi 5932	. . . . . 6 ⊢ ran (card ↾ ran 𝑅1) ⊆ ran card
14		frn 6717	. . . . . . 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 3947	. . . . 5 ⊢ ran (card ↾ ran 𝑅1) ⊆ On
17	11, 16	eqsstri 3984	. . . 4 ⊢ ran (card ∘ 𝑅1) ⊆ On
18		df-f 6544	. . . 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 6258	. . . 4 ⊢ dom (card ∘ 𝑅1) = (◡𝑅1 “ dom card)
21	20	feq2i 6701	. . 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 7057	. . . . . . . . 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 6389	. . . . . . 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 9756	. . . . . . . . 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 10035	. . . . . . 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 7057	. . . . . . 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 3207	. . . 4 ⊢ ∀𝑥 ∈ (◡𝑅1 “ dom card)∀𝑦 ∈ 𝑥 𝑦 ∈ (◡𝑅1 “ dom card)
39		dftr5 5224	. . . 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 6086	. . . . 5 ⊢ (◡𝑅1 “ dom card) ⊆ dom 𝑅1
42		dffn2 6711	. . . . . . 7 ⊢ (𝑅1 Fn On ↔ 𝑅1:On⟶V)
43	4, 42	mpbi 233	. . . . . 6 ⊢ 𝑅1:On⟶V
44	43	fdmi 6721	. . . . 5 ⊢ dom 𝑅1 = On
45	41, 44	sseqtri 3986	. . . 4 ⊢ (◡𝑅1 “ dom card) ⊆ On
46		epweon 7776	. . . 4 ⊢ E We On
47		wess 5649	. . . 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 6367	. . 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 9749	. . . . . . 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 9986	. . . . . . 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 6982	. . . . . 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 6982	. . . . . 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 2880	. . . 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 8337	1 ⊢ Smo (card ∘ 𝑅1)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906   class class class wbr 5111  Tr wtr 5220   E cep 5562   We wwe 5615  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666   ∘ ccom 5667  Ord word 6363  Oncon0 6364  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  Smo wsmo 8334   ≈ cen 8942   ≺ csdm 8944  𝑅₁cr1 9737  cardccrd 9933

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-om 7865  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-smo 8335  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-r1 9739  df-card 9937

This theorem is used by: (None)