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Theorem smobeth 10006
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 9369 . . . . . . 7 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 ffun 6506 . . . . . . 7 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → Fun card)
31, 2ax-mp 5 . . . . . 6 Fun card
4 r1fnon 9193 . . . . . . 7 𝑅1 Fn On
5 fnfun 6441 . . . . . . 7 (𝑅1 Fn On → Fun 𝑅1)
64, 5ax-mp 5 . . . . . 6 Fun 𝑅1
7 funco 6383 . . . . . 6 ((Fun card ∧ Fun 𝑅1) → Fun (card ∘ 𝑅1))
83, 6, 7mp2an 691 . . . . 5 Fun (card ∘ 𝑅1)
9 funfn 6373 . . . . 5 (Fun (card ∘ 𝑅1) ↔ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1))
108, 9mpbi 233 . . . 4 (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1)
11 rnco 6092 . . . . 5 ran (card ∘ 𝑅1) = ran (card ↾ ran 𝑅1)
12 resss 5865 . . . . . . 7 (card ↾ ran 𝑅1) ⊆ card
1312rnssi 5797 . . . . . 6 ran (card ↾ ran 𝑅1) ⊆ ran card
14 frn 6509 . . . . . . 7 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → ran card ⊆ On)
151, 14ax-mp 5 . . . . . 6 ran card ⊆ On
1613, 15sstri 3962 . . . . 5 ran (card ↾ ran 𝑅1) ⊆ On
1711, 16eqsstri 3987 . . . 4 ran (card ∘ 𝑅1) ⊆ On
18 df-f 6347 . . . 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 6094 . . . 4 dom (card ∘ 𝑅1) = (𝑅1 “ dom card)
2120feq2i 6495 . . 3 ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ (card ∘ 𝑅1):(𝑅1 “ dom card)⟶On)
2219, 21mpbi 233 . 2 (card ∘ 𝑅1):(𝑅1 “ dom card)⟶On
23 elpreima 6819 . . . . . . . . 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 501 . . . . . . 7 (𝑥 ∈ (𝑅1 “ dom card) → 𝑥 ∈ On)
26 onelon 6203 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
2725, 26sylan 583 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑦 ∈ On)
2824simprbi 500 . . . . . . . 8 (𝑥 ∈ (𝑅1 “ dom card) → (𝑅1𝑥) ∈ dom card)
2928adantr 484 . . . . . . 7 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑥) ∈ dom card)
30 r1ord2 9207 . . . . . . . . 9 (𝑥 ∈ On → (𝑦𝑥 → (𝑅1𝑦) ⊆ (𝑅1𝑥)))
3130imp 410 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ⊆ (𝑅1𝑥))
3225, 31sylan 583 . . . . . . 7 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ⊆ (𝑅1𝑥))
33 ssnum 9463 . . . . . . 7 (((𝑅1𝑥) ∈ dom card ∧ (𝑅1𝑦) ⊆ (𝑅1𝑥)) → (𝑅1𝑦) ∈ dom card)
3429, 32, 33syl2anc 587 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ∈ dom card)
35 elpreima 6819 . . . . . . 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 586 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑦 ∈ (𝑅1 “ dom card))
3837rgen2 3198 . . . 4 𝑥 ∈ (𝑅1 “ dom card)∀𝑦𝑥 𝑦 ∈ (𝑅1 “ dom card)
39 dftr5 5161 . . . 4 (Tr (𝑅1 “ dom card) ↔ ∀𝑥 ∈ (𝑅1 “ dom card)∀𝑦𝑥 𝑦 ∈ (𝑅1 “ dom card))
4038, 39mpbir 234 . . 3 Tr (𝑅1 “ dom card)
41 cnvimass 5936 . . . . 5 (𝑅1 “ dom card) ⊆ dom 𝑅1
42 dffn2 6505 . . . . . . 7 (𝑅1 Fn On ↔ 𝑅1:On⟶V)
434, 42mpbi 233 . . . . . 6 𝑅1:On⟶V
4443fdmi 6514 . . . . 5 dom 𝑅1 = On
4541, 44sseqtri 3989 . . . 4 (𝑅1 “ dom card) ⊆ On
46 epweon 7491 . . . 4 E We On
47 wess 5529 . . . 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 6181 . . 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 9200 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
5225, 51sylan 583 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
53 cardsdom2 9414 . . . . . . 7 (((𝑅1𝑦) ∈ dom card ∧ (𝑅1𝑥) ∈ dom card) → ((card‘(𝑅1𝑦)) ∈ (card‘(𝑅1𝑥)) ↔ (𝑅1𝑦) ≺ (𝑅1𝑥)))
5434, 29, 53syl2anc 587 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card‘(𝑅1𝑦)) ∈ (card‘(𝑅1𝑥)) ↔ (𝑅1𝑦) ≺ (𝑅1𝑥)))
5552, 54mpbird 260 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (card‘(𝑅1𝑦)) ∈ (card‘(𝑅1𝑥)))
56 fvco2 6749 . . . . . 6 ((𝑅1 Fn On ∧ 𝑦 ∈ On) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1𝑦)))
574, 27, 56sylancr 590 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1𝑦)))
5825adantr 484 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑥 ∈ On)
59 fvco2 6749 . . . . . 6 ((𝑅1 Fn On ∧ 𝑥 ∈ On) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1𝑥)))
604, 58, 59sylancr 590 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1𝑥)))
6155, 57, 603eltr4d 2931 . . . 4 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥))
6261ex 416 . . 3 (𝑥 ∈ (𝑅1 “ dom card) → (𝑦𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
6362adantl 485 . 2 ((𝑦 ∈ (𝑅1 “ dom card) ∧ 𝑥 ∈ (𝑅1 “ dom card)) → (𝑦𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
6422, 50, 63, 20issmo 7981 1 Smo (card ∘ 𝑅1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2802  wral 3133  wrex 3134  Vcvv 3480  wss 3919   class class class wbr 5052  Tr wtr 5158   E cep 5451   We wwe 5500  ccnv 5541  dom cdm 5542  ran crn 5543  cres 5544  cima 5545  ccom 5546  Ord word 6177  Oncon0 6178  Fun wfun 6337   Fn wfn 6338  wf 6339  cfv 6343  Smo wsmo 7978  cen 8502  csdm 8504  𝑅1cr1 9188  cardccrd 9361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-om 7575  df-wrecs 7943  df-smo 7979  df-recs 8004  df-rdg 8042  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-r1 9190  df-card 9365
This theorem is referenced by: (None)
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