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Theorem smobeth 10626
Description: The beth function is strictly monotone. This function is not strictly the beth function, but rather bethA 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.
StepHypRef Expression
1 cardf2 9983 . . . . . . 7 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 ffun 6739 . . . . . . 7 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → Fun card)
31, 2ax-mp 5 . . . . . 6 Fun card
4 r1fnon 9807 . . . . . . 7 𝑅1 Fn On
5 fnfun 6668 . . . . . . 7 (𝑅1 Fn On → Fun 𝑅1)
64, 5ax-mp 5 . . . . . 6 Fun 𝑅1
7 funco 6606 . . . . . 6 ((Fun card ∧ Fun 𝑅1) → Fun (card ∘ 𝑅1))
83, 6, 7mp2an 692 . . . . 5 Fun (card ∘ 𝑅1)
9 funfn 6596 . . . . 5 (Fun (card ∘ 𝑅1) ↔ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1))
108, 9mpbi 230 . . . 4 (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1)
11 rnco 6272 . . . . 5 ran (card ∘ 𝑅1) = ran (card ↾ ran 𝑅1)
12 resss 6019 . . . . . . 7 (card ↾ ran 𝑅1) ⊆ card
1312rnssi 5951 . . . . . 6 ran (card ↾ ran 𝑅1) ⊆ ran card
14 frn 6743 . . . . . . 7 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → ran card ⊆ On)
151, 14ax-mp 5 . . . . . 6 ran card ⊆ On
1613, 15sstri 3993 . . . . 5 ran (card ↾ ran 𝑅1) ⊆ On
1711, 16eqsstri 4030 . . . 4 ran (card ∘ 𝑅1) ⊆ On
18 df-f 6565 . . . 4 ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ ((card ∘ 𝑅1) Fn dom (card ∘ 𝑅1) ∧ ran (card ∘ 𝑅1) ⊆ On))
1910, 17, 18mpbir2an 711 . . 3 (card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On
20 dmco 6274 . . . 4 dom (card ∘ 𝑅1) = (𝑅1 “ dom card)
2120feq2i 6728 . . 3 ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ (card ∘ 𝑅1):(𝑅1 “ dom card)⟶On)
2219, 21mpbi 230 . 2 (card ∘ 𝑅1):(𝑅1 “ dom card)⟶On
23 elpreima 7078 . . . . . . . . 9 (𝑅1 Fn On → (𝑥 ∈ (𝑅1 “ dom card) ↔ (𝑥 ∈ On ∧ (𝑅1𝑥) ∈ dom card)))
244, 23ax-mp 5 . . . . . . . 8 (𝑥 ∈ (𝑅1 “ dom card) ↔ (𝑥 ∈ On ∧ (𝑅1𝑥) ∈ dom card))
2524simplbi 497 . . . . . . 7 (𝑥 ∈ (𝑅1 “ dom card) → 𝑥 ∈ On)
26 onelon 6409 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
2725, 26sylan 580 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑦 ∈ On)
2824simprbi 496 . . . . . . . 8 (𝑥 ∈ (𝑅1 “ dom card) → (𝑅1𝑥) ∈ dom card)
2928adantr 480 . . . . . . 7 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑥) ∈ dom card)
30 r1ord2 9821 . . . . . . . . 9 (𝑥 ∈ On → (𝑦𝑥 → (𝑅1𝑦) ⊆ (𝑅1𝑥)))
3130imp 406 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ⊆ (𝑅1𝑥))
3225, 31sylan 580 . . . . . . 7 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ⊆ (𝑅1𝑥))
33 ssnum 10079 . . . . . . 7 (((𝑅1𝑥) ∈ dom card ∧ (𝑅1𝑦) ⊆ (𝑅1𝑥)) → (𝑅1𝑦) ∈ dom card)
3429, 32, 33syl2anc 584 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ∈ dom card)
35 elpreima 7078 . . . . . . 7 (𝑅1 Fn On → (𝑦 ∈ (𝑅1 “ dom card) ↔ (𝑦 ∈ On ∧ (𝑅1𝑦) ∈ dom card)))
364, 35ax-mp 5 . . . . . 6 (𝑦 ∈ (𝑅1 “ dom card) ↔ (𝑦 ∈ On ∧ (𝑅1𝑦) ∈ dom card))
3727, 34, 36sylanbrc 583 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑦 ∈ (𝑅1 “ dom card))
3837rgen2 3199 . . . 4 𝑥 ∈ (𝑅1 “ dom card)∀𝑦𝑥 𝑦 ∈ (𝑅1 “ dom card)
39 dftr5 5263 . . . 4 (Tr (𝑅1 “ dom card) ↔ ∀𝑥 ∈ (𝑅1 “ dom card)∀𝑦𝑥 𝑦 ∈ (𝑅1 “ dom card))
4038, 39mpbir 231 . . 3 Tr (𝑅1 “ dom card)
41 cnvimass 6100 . . . . 5 (𝑅1 “ dom card) ⊆ dom 𝑅1
42 dffn2 6738 . . . . . . 7 (𝑅1 Fn On ↔ 𝑅1:On⟶V)
434, 42mpbi 230 . . . . . 6 𝑅1:On⟶V
4443fdmi 6747 . . . . 5 dom 𝑅1 = On
4541, 44sseqtri 4032 . . . 4 (𝑅1 “ dom card) ⊆ On
46 epweon 7795 . . . 4 E We On
47 wess 5671 . . . 4 ((𝑅1 “ dom card) ⊆ On → ( E We On → E We (𝑅1 “ dom card)))
4845, 46, 47mp2 9 . . 3 E We (𝑅1 “ dom card)
49 df-ord 6387 . . 3 (Ord (𝑅1 “ dom card) ↔ (Tr (𝑅1 “ dom card) ∧ E We (𝑅1 “ dom card)))
5040, 48, 49mpbir2an 711 . 2 Ord (𝑅1 “ dom card)
51 r1sdom 9814 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
5225, 51sylan 580 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
53 cardsdom2 10028 . . . . . . 7 (((𝑅1𝑦) ∈ dom card ∧ (𝑅1𝑥) ∈ dom card) → ((card‘(𝑅1𝑦)) ∈ (card‘(𝑅1𝑥)) ↔ (𝑅1𝑦) ≺ (𝑅1𝑥)))
5434, 29, 53syl2anc 584 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card‘(𝑅1𝑦)) ∈ (card‘(𝑅1𝑥)) ↔ (𝑅1𝑦) ≺ (𝑅1𝑥)))
5552, 54mpbird 257 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (card‘(𝑅1𝑦)) ∈ (card‘(𝑅1𝑥)))
56 fvco2 7006 . . . . . 6 ((𝑅1 Fn On ∧ 𝑦 ∈ On) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1𝑦)))
574, 27, 56sylancr 587 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1𝑦)))
5825adantr 480 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑥 ∈ On)
59 fvco2 7006 . . . . . 6 ((𝑅1 Fn On ∧ 𝑥 ∈ On) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1𝑥)))
604, 58, 59sylancr 587 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1𝑥)))
6155, 57, 603eltr4d 2856 . . . 4 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥))
6261ex 412 . . 3 (𝑥 ∈ (𝑅1 “ dom card) → (𝑦𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
6362adantl 481 . 2 ((𝑦 ∈ (𝑅1 “ dom card) ∧ 𝑥 ∈ (𝑅1 “ dom card)) → (𝑦𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
6422, 50, 63, 20issmo 8388 1 Smo (card ∘ 𝑅1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  wss 3951   class class class wbr 5143  Tr wtr 5259   E cep 5583   We wwe 5636  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  ccom 5689  Ord word 6383  Oncon0 6384  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  Smo wsmo 8385  cen 8982  csdm 8984  𝑅1cr1 9802  cardccrd 9975
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-smo 8386  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-r1 9804  df-card 9979
This theorem is referenced by: (None)
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