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Theorem smobeth 9609
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 8968 . . . . . . 7 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 ffun 6188 . . . . . . 7 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → Fun card)
31, 2ax-mp 5 . . . . . 6 Fun card
4 r1fnon 8793 . . . . . . 7 𝑅1 Fn On
5 fnfun 6128 . . . . . . 7 (𝑅1 Fn On → Fun 𝑅1)
64, 5ax-mp 5 . . . . . 6 Fun 𝑅1
7 funco 6071 . . . . . 6 ((Fun card ∧ Fun 𝑅1) → Fun (card ∘ 𝑅1))
83, 6, 7mp2an 664 . . . . 5 Fun (card ∘ 𝑅1)
9 funfn 6061 . . . . 5 (Fun (card ∘ 𝑅1) ↔ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1))
108, 9mpbi 220 . . . 4 (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1)
11 rnco 5785 . . . . 5 ran (card ∘ 𝑅1) = ran (card ↾ ran 𝑅1)
12 resss 5563 . . . . . . 7 (card ↾ ran 𝑅1) ⊆ card
13 rnss 5492 . . . . . . 7 ((card ↾ ran 𝑅1) ⊆ card → ran (card ↾ ran 𝑅1) ⊆ ran card)
1412, 13ax-mp 5 . . . . . 6 ran (card ↾ ran 𝑅1) ⊆ ran card
15 frn 6193 . . . . . . 7 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → ran card ⊆ On)
161, 15ax-mp 5 . . . . . 6 ran card ⊆ On
1714, 16sstri 3759 . . . . 5 ran (card ↾ ran 𝑅1) ⊆ On
1811, 17eqsstri 3782 . . . 4 ran (card ∘ 𝑅1) ⊆ On
19 df-f 6035 . . . 4 ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ ((card ∘ 𝑅1) Fn dom (card ∘ 𝑅1) ∧ ran (card ∘ 𝑅1) ⊆ On))
2010, 18, 19mpbir2an 682 . . 3 (card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On
21 dmco 5787 . . . 4 dom (card ∘ 𝑅1) = (𝑅1 “ dom card)
2221feq2i 6177 . . 3 ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ (card ∘ 𝑅1):(𝑅1 “ dom card)⟶On)
2320, 22mpbi 220 . 2 (card ∘ 𝑅1):(𝑅1 “ dom card)⟶On
24 elpreima 6480 . . . . . . . . 9 (𝑅1 Fn On → (𝑥 ∈ (𝑅1 “ dom card) ↔ (𝑥 ∈ On ∧ (𝑅1𝑥) ∈ dom card)))
254, 24ax-mp 5 . . . . . . . 8 (𝑥 ∈ (𝑅1 “ dom card) ↔ (𝑥 ∈ On ∧ (𝑅1𝑥) ∈ dom card))
2625simplbi 479 . . . . . . 7 (𝑥 ∈ (𝑅1 “ dom card) → 𝑥 ∈ On)
27 onelon 5891 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
2826, 27sylan 561 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑦 ∈ On)
2925simprbi 478 . . . . . . . 8 (𝑥 ∈ (𝑅1 “ dom card) → (𝑅1𝑥) ∈ dom card)
3029adantr 466 . . . . . . 7 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑥) ∈ dom card)
31 r1ord2 8807 . . . . . . . . 9 (𝑥 ∈ On → (𝑦𝑥 → (𝑅1𝑦) ⊆ (𝑅1𝑥)))
3231imp 393 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ⊆ (𝑅1𝑥))
3326, 32sylan 561 . . . . . . 7 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ⊆ (𝑅1𝑥))
34 ssnum 9061 . . . . . . 7 (((𝑅1𝑥) ∈ dom card ∧ (𝑅1𝑦) ⊆ (𝑅1𝑥)) → (𝑅1𝑦) ∈ dom card)
3530, 33, 34syl2anc 565 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ∈ dom card)
36 elpreima 6480 . . . . . . 7 (𝑅1 Fn On → (𝑦 ∈ (𝑅1 “ dom card) ↔ (𝑦 ∈ On ∧ (𝑅1𝑦) ∈ dom card)))
374, 36ax-mp 5 . . . . . 6 (𝑦 ∈ (𝑅1 “ dom card) ↔ (𝑦 ∈ On ∧ (𝑅1𝑦) ∈ dom card))
3828, 35, 37sylanbrc 564 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑦 ∈ (𝑅1 “ dom card))
3938rgen2 3123 . . . 4 𝑥 ∈ (𝑅1 “ dom card)∀𝑦𝑥 𝑦 ∈ (𝑅1 “ dom card)
40 dftr5 4887 . . . 4 (Tr (𝑅1 “ dom card) ↔ ∀𝑥 ∈ (𝑅1 “ dom card)∀𝑦𝑥 𝑦 ∈ (𝑅1 “ dom card))
4139, 40mpbir 221 . . 3 Tr (𝑅1 “ dom card)
42 cnvimass 5626 . . . . 5 (𝑅1 “ dom card) ⊆ dom 𝑅1
43 dffn2 6187 . . . . . . 7 (𝑅1 Fn On ↔ 𝑅1:On⟶V)
444, 43mpbi 220 . . . . . 6 𝑅1:On⟶V
4544fdmi 6192 . . . . 5 dom 𝑅1 = On
4642, 45sseqtri 3784 . . . 4 (𝑅1 “ dom card) ⊆ On
47 epweon 7129 . . . 4 E We On
48 wess 5236 . . . 4 ((𝑅1 “ dom card) ⊆ On → ( E We On → E We (𝑅1 “ dom card)))
4946, 47, 48mp2 9 . . 3 E We (𝑅1 “ dom card)
50 df-ord 5869 . . 3 (Ord (𝑅1 “ dom card) ↔ (Tr (𝑅1 “ dom card) ∧ E We (𝑅1 “ dom card)))
5141, 49, 50mpbir2an 682 . 2 Ord (𝑅1 “ dom card)
52 r1sdom 8800 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
5326, 52sylan 561 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
54 cardsdom2 9013 . . . . . . 7 (((𝑅1𝑦) ∈ dom card ∧ (𝑅1𝑥) ∈ dom card) → ((card‘(𝑅1𝑦)) ∈ (card‘(𝑅1𝑥)) ↔ (𝑅1𝑦) ≺ (𝑅1𝑥)))
5535, 30, 54syl2anc 565 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card‘(𝑅1𝑦)) ∈ (card‘(𝑅1𝑥)) ↔ (𝑅1𝑦) ≺ (𝑅1𝑥)))
5653, 55mpbird 247 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (card‘(𝑅1𝑦)) ∈ (card‘(𝑅1𝑥)))
57 fvco2 6415 . . . . . 6 ((𝑅1 Fn On ∧ 𝑦 ∈ On) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1𝑦)))
584, 28, 57sylancr 567 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1𝑦)))
5926adantr 466 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑥 ∈ On)
60 fvco2 6415 . . . . . 6 ((𝑅1 Fn On ∧ 𝑥 ∈ On) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1𝑥)))
614, 59, 60sylancr 567 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1𝑥)))
6256, 58, 613eltr4d 2864 . . . 4 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥))
6362ex 397 . . 3 (𝑥 ∈ (𝑅1 “ dom card) → (𝑦𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
6463adantl 467 . 2 ((𝑦 ∈ (𝑅1 “ dom card) ∧ 𝑥 ∈ (𝑅1 “ dom card)) → (𝑦𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
6523, 51, 64, 21issmo 7597 1 Smo (card ∘ 𝑅1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  {cab 2756  wral 3060  wrex 3061  Vcvv 3349  wss 3721   class class class wbr 4784  Tr wtr 4884   E cep 5161   We wwe 5207  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  ccom 5253  Ord word 5865  Oncon0 5866  Fun wfun 6025   Fn wfn 6026  wf 6027  cfv 6031  Smo wsmo 7594  cen 8105  csdm 8107  𝑅1cr1 8788  cardccrd 8960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-om 7212  df-wrecs 7558  df-smo 7595  df-recs 7620  df-rdg 7658  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-r1 8790  df-card 8964
This theorem is referenced by: (None)
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