Theorem smobeth 10273

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 9632	. . . . . . 7 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
2		ffun 6587	. . . . . . 7 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On → Fun card)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun card
4		r1fnon 9456	. . . . . . 7 ⊢ 𝑅1 Fn On
5		fnfun 6517	. . . . . . 7 ⊢ (𝑅1 Fn On → Fun 𝑅1)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ Fun 𝑅1
7		funco 6458	. . . . . 6 ⊢ ((Fun card ∧ Fun 𝑅1) → Fun (card ∘ 𝑅1))
8	3, 6, 7	mp2an 688	. . . . 5 ⊢ Fun (card ∘ 𝑅1)
9		funfn 6448	. . . . 5 ⊢ (Fun (card ∘ 𝑅1) ↔ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1))
10	8, 9	mpbi 229	. . . 4 ⊢ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1)
11		rnco 6145	. . . . 5 ⊢ ran (card ∘ 𝑅1) = ran (card ↾ ran 𝑅1)
12		resss 5905	. . . . . . 7 ⊢ (card ↾ ran 𝑅1) ⊆ card
13	12	rnssi 5838	. . . . . 6 ⊢ ran (card ↾ ran 𝑅1) ⊆ ran card
14		frn 6591	. . . . . . 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 3926	. . . . 5 ⊢ ran (card ↾ ran 𝑅1) ⊆ On
17	11, 16	eqsstri 3951	. . . 4 ⊢ ran (card ∘ 𝑅1) ⊆ On
18		df-f 6422	. . . 4 ⊢ ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ ((card ∘ 𝑅1) Fn dom (card ∘ 𝑅1) ∧ ran (card ∘ 𝑅1) ⊆ On))
19	10, 17, 18	mpbir2an 707	. . 3 ⊢ (card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On
20		dmco 6147	. . . 4 ⊢ dom (card ∘ 𝑅1) = (◡𝑅1 “ dom card)
21	20	feq2i 6576	. . 3 ⊢ ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ (card ∘ 𝑅1):(◡𝑅1 “ dom card)⟶On)
22	19, 21	mpbi 229	. 2 ⊢ (card ∘ 𝑅1):(◡𝑅1 “ dom card)⟶On
23		elpreima 6917	. . . . . . . . 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 6276	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
27	25, 26	sylan 579	. . . . . 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 9470	. . . . . . . . 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 579	. . . . . . 7 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥))
33		ssnum 9726	. . . . . . 7 ⊢ (((𝑅1‘𝑥) ∈ dom card ∧ (𝑅1‘𝑦) ⊆ (𝑅1‘𝑥)) → (𝑅1‘𝑦) ∈ dom card)
34	29, 32, 33	syl2anc 583	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ∈ dom card)
35		elpreima 6917	. . . . . . 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 582	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (◡𝑅1 “ dom card))
38	37	rgen2 3126	. . . 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 230	. . 3 ⊢ Tr (◡𝑅1 “ dom card)
41		cnvimass 5978	. . . . 5 ⊢ (◡𝑅1 “ dom card) ⊆ dom 𝑅1
42		dffn2 6586	. . . . . . 7 ⊢ (𝑅1 Fn On ↔ 𝑅1:On⟶V)
43	4, 42	mpbi 229	. . . . . 6 ⊢ 𝑅1:On⟶V
44	43	fdmi 6596	. . . . 5 ⊢ dom 𝑅1 = On
45	41, 44	sseqtri 3953	. . . 4 ⊢ (◡𝑅1 “ dom card) ⊆ On
46		epweon 7603	. . . 4 ⊢ E We On
47		wess 5567	. . . 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 6254	. . 3 ⊢ (Ord (◡𝑅1 “ dom card) ↔ (Tr (◡𝑅1 “ dom card) ∧ E We (◡𝑅1 “ dom card)))
50	40, 48, 49	mpbir2an 707	. 2 ⊢ Ord (◡𝑅1 “ dom card)
51		r1sdom 9463	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
52	25, 51	sylan 579	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (𝑅1‘𝑦) ≺ (𝑅1‘𝑥))
53		cardsdom2 9677	. . . . . . 7 ⊢ (((𝑅1‘𝑦) ∈ dom card ∧ (𝑅1‘𝑥) ∈ dom card) → ((card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑥)))
54	34, 29, 53	syl2anc 583	. . . . . 6 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)) ↔ (𝑅1‘𝑦) ≺ (𝑅1‘𝑥)))
55	52, 54	mpbird 256	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → (card‘(𝑅1‘𝑦)) ∈ (card‘(𝑅1‘𝑥)))
56		fvco2 6847	. . . . . 6 ⊢ ((𝑅1 Fn On ∧ 𝑦 ∈ On) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1‘𝑦)))
57	4, 27, 56	sylancr 586	. . . . 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 6847	. . . . . 6 ⊢ ((𝑅1 Fn On ∧ 𝑥 ∈ On) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1‘𝑥)))
60	4, 58, 59	sylancr 586	. . . . 5 ⊢ ((𝑥 ∈ (◡𝑅1 “ dom card) ∧ 𝑦 ∈ 𝑥) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1‘𝑥)))
61	55, 57, 60	3eltr4d 2854	. . . 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 8150	1 ⊢ Smo (card ∘ 𝑅1)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883   class class class wbr 5070  Tr wtr 5187   E cep 5485   We wwe 5534  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583   ∘ ccom 5584  Ord word 6250  Oncon0 6251  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  Smo wsmo 8147   ≈ cen 8688   ≺ csdm 8690  𝑅₁cr1 9451  cardccrd 9624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-smo 8148  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-r1 9453  df-card 9628

This theorem is referenced by: (None)