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Theorem smobeth 10610
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 9967 . . . . . . 7 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 ffun 6725 . . . . . . 7 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → Fun card)
31, 2ax-mp 5 . . . . . 6 Fun card
4 r1fnon 9791 . . . . . . 7 𝑅1 Fn On
5 fnfun 6654 . . . . . . 7 (𝑅1 Fn On → Fun 𝑅1)
64, 5ax-mp 5 . . . . . 6 Fun 𝑅1
7 funco 6593 . . . . . 6 ((Fun card ∧ Fun 𝑅1) → Fun (card ∘ 𝑅1))
83, 6, 7mp2an 691 . . . . 5 Fun (card ∘ 𝑅1)
9 funfn 6583 . . . . 5 (Fun (card ∘ 𝑅1) ↔ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1))
108, 9mpbi 229 . . . 4 (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1)
11 rnco 6256 . . . . 5 ran (card ∘ 𝑅1) = ran (card ↾ ran 𝑅1)
12 resss 6010 . . . . . . 7 (card ↾ ran 𝑅1) ⊆ card
1312rnssi 5942 . . . . . 6 ran (card ↾ ran 𝑅1) ⊆ ran card
14 frn 6729 . . . . . . 7 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → ran card ⊆ On)
151, 14ax-mp 5 . . . . . 6 ran card ⊆ On
1613, 15sstri 3989 . . . . 5 ran (card ↾ ran 𝑅1) ⊆ On
1711, 16eqsstri 4014 . . . 4 ran (card ∘ 𝑅1) ⊆ On
18 df-f 6552 . . . 4 ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ ((card ∘ 𝑅1) Fn dom (card ∘ 𝑅1) ∧ ran (card ∘ 𝑅1) ⊆ On))
1910, 17, 18mpbir2an 710 . . 3 (card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On
20 dmco 6258 . . . 4 dom (card ∘ 𝑅1) = (𝑅1 “ dom card)
2120feq2i 6714 . . 3 ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ (card ∘ 𝑅1):(𝑅1 “ dom card)⟶On)
2219, 21mpbi 229 . 2 (card ∘ 𝑅1):(𝑅1 “ dom card)⟶On
23 elpreima 7067 . . . . . . . . 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 6394 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
2725, 26sylan 579 . . . . . 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 9805 . . . . . . . . 9 (𝑥 ∈ On → (𝑦𝑥 → (𝑅1𝑦) ⊆ (𝑅1𝑥)))
3130imp 406 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ⊆ (𝑅1𝑥))
3225, 31sylan 579 . . . . . . 7 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ⊆ (𝑅1𝑥))
33 ssnum 10063 . . . . . . 7 (((𝑅1𝑥) ∈ dom card ∧ (𝑅1𝑦) ⊆ (𝑅1𝑥)) → (𝑅1𝑦) ∈ dom card)
3429, 32, 33syl2anc 583 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ∈ dom card)
35 elpreima 7067 . . . . . . 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 582 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑦 ∈ (𝑅1 “ dom card))
3837rgen2 3194 . . . 4 𝑥 ∈ (𝑅1 “ dom card)∀𝑦𝑥 𝑦 ∈ (𝑅1 “ dom card)
39 dftr5 5269 . . . 4 (Tr (𝑅1 “ dom card) ↔ ∀𝑥 ∈ (𝑅1 “ dom card)∀𝑦𝑥 𝑦 ∈ (𝑅1 “ dom card))
4038, 39mpbir 230 . . 3 Tr (𝑅1 “ dom card)
41 cnvimass 6085 . . . . 5 (𝑅1 “ dom card) ⊆ dom 𝑅1
42 dffn2 6724 . . . . . . 7 (𝑅1 Fn On ↔ 𝑅1:On⟶V)
434, 42mpbi 229 . . . . . 6 𝑅1:On⟶V
4443fdmi 6734 . . . . 5 dom 𝑅1 = On
4541, 44sseqtri 4016 . . . 4 (𝑅1 “ dom card) ⊆ On
46 epweon 7777 . . . 4 E We On
47 wess 5665 . . . 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 6372 . . 3 (Ord (𝑅1 “ dom card) ↔ (Tr (𝑅1 “ dom card) ∧ E We (𝑅1 “ dom card)))
5040, 48, 49mpbir2an 710 . 2 Ord (𝑅1 “ dom card)
51 r1sdom 9798 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
5225, 51sylan 579 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
53 cardsdom2 10012 . . . . . . 7 (((𝑅1𝑦) ∈ dom card ∧ (𝑅1𝑥) ∈ dom card) → ((card‘(𝑅1𝑦)) ∈ (card‘(𝑅1𝑥)) ↔ (𝑅1𝑦) ≺ (𝑅1𝑥)))
5434, 29, 53syl2anc 583 . . . . . 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 6995 . . . . . 6 ((𝑅1 Fn On ∧ 𝑦 ∈ On) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1𝑦)))
574, 27, 56sylancr 586 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1𝑦)))
5825adantr 480 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑥 ∈ On)
59 fvco2 6995 . . . . . 6 ((𝑅1 Fn On ∧ 𝑥 ∈ On) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1𝑥)))
604, 58, 59sylancr 586 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1𝑥)))
6155, 57, 603eltr4d 2844 . . . 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 8369 1 Smo (card ∘ 𝑅1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  {cab 2705  wral 3058  wrex 3067  Vcvv 3471  wss 3947   class class class wbr 5148  Tr wtr 5265   E cep 5581   We wwe 5632  ccnv 5677  dom cdm 5678  ran crn 5679  cres 5680  cima 5681  ccom 5682  Ord word 6368  Oncon0 6369  Fun wfun 6542   Fn wfn 6543  wf 6544  cfv 6548  Smo wsmo 8366  cen 8961  csdm 8963  𝑅1cr1 9786  cardccrd 9959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-smo 8367  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-r1 9788  df-card 9963
This theorem is referenced by: (None)
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