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Theorem smobeth 9405
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 8766 . . . . . . 7 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
2 ffun 6046 . . . . . . 7 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → Fun card)
31, 2ax-mp 5 . . . . . 6 Fun card
4 r1fnon 8627 . . . . . . 7 𝑅1 Fn On
5 fnfun 5986 . . . . . . 7 (𝑅1 Fn On → Fun 𝑅1)
64, 5ax-mp 5 . . . . . 6 Fun 𝑅1
7 funco 5926 . . . . . 6 ((Fun card ∧ Fun 𝑅1) → Fun (card ∘ 𝑅1))
83, 6, 7mp2an 708 . . . . 5 Fun (card ∘ 𝑅1)
9 funfn 5916 . . . . 5 (Fun (card ∘ 𝑅1) ↔ (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1))
108, 9mpbi 220 . . . 4 (card ∘ 𝑅1) Fn dom (card ∘ 𝑅1)
11 rnco 5639 . . . . 5 ran (card ∘ 𝑅1) = ran (card ↾ ran 𝑅1)
12 resss 5420 . . . . . . 7 (card ↾ ran 𝑅1) ⊆ card
13 rnss 5352 . . . . . . 7 ((card ↾ ran 𝑅1) ⊆ card → ran (card ↾ ran 𝑅1) ⊆ ran card)
1412, 13ax-mp 5 . . . . . 6 ran (card ↾ ran 𝑅1) ⊆ ran card
15 frn 6051 . . . . . . 7 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On → ran card ⊆ On)
161, 15ax-mp 5 . . . . . 6 ran card ⊆ On
1714, 16sstri 3610 . . . . 5 ran (card ↾ ran 𝑅1) ⊆ On
1811, 17eqsstri 3633 . . . 4 ran (card ∘ 𝑅1) ⊆ On
19 df-f 5890 . . . 4 ((card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On ↔ ((card ∘ 𝑅1) Fn dom (card ∘ 𝑅1) ∧ ran (card ∘ 𝑅1) ⊆ On))
2010, 18, 19mpbir2an 955 . . 3 (card ∘ 𝑅1):dom (card ∘ 𝑅1)⟶On
21 dmco 5641 . . . 4 dom (card ∘ 𝑅1) = (𝑅1 “ dom card)
2221feq2i 6035 . . 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 6335 . . . . . . . . 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 476 . . . . . . 7 (𝑥 ∈ (𝑅1 “ dom card) → 𝑥 ∈ On)
27 onelon 5746 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
2826, 27sylan 488 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑦 ∈ On)
2925simprbi 480 . . . . . . . 8 (𝑥 ∈ (𝑅1 “ dom card) → (𝑅1𝑥) ∈ dom card)
3029adantr 481 . . . . . . 7 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑥) ∈ dom card)
31 r1ord2 8641 . . . . . . . . 9 (𝑥 ∈ On → (𝑦𝑥 → (𝑅1𝑦) ⊆ (𝑅1𝑥)))
3231imp 445 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ⊆ (𝑅1𝑥))
3326, 32sylan 488 . . . . . . 7 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ⊆ (𝑅1𝑥))
34 ssnum 8859 . . . . . . 7 (((𝑅1𝑥) ∈ dom card ∧ (𝑅1𝑦) ⊆ (𝑅1𝑥)) → (𝑅1𝑦) ∈ dom card)
3530, 33, 34syl2anc 693 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ∈ dom card)
36 elpreima 6335 . . . . . . 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 698 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑦 ∈ (𝑅1 “ dom card))
3938rgen2 2974 . . . 4 𝑥 ∈ (𝑅1 “ dom card)∀𝑦𝑥 𝑦 ∈ (𝑅1 “ dom card)
40 dftr5 4753 . . . 4 (Tr (𝑅1 “ dom card) ↔ ∀𝑥 ∈ (𝑅1 “ dom card)∀𝑦𝑥 𝑦 ∈ (𝑅1 “ dom card))
4139, 40mpbir 221 . . 3 Tr (𝑅1 “ dom card)
42 cnvimass 5483 . . . . 5 (𝑅1 “ dom card) ⊆ dom 𝑅1
43 dffn2 6045 . . . . . . 7 (𝑅1 Fn On ↔ 𝑅1:On⟶V)
444, 43mpbi 220 . . . . . 6 𝑅1:On⟶V
4544fdmi 6050 . . . . 5 dom 𝑅1 = On
4642, 45sseqtri 3635 . . . 4 (𝑅1 “ dom card) ⊆ On
47 epweon 6980 . . . 4 E We On
48 wess 5099 . . . 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 5724 . . 3 (Ord (𝑅1 “ dom card) ↔ (Tr (𝑅1 “ dom card) ∧ E We (𝑅1 “ dom card)))
5141, 49, 50mpbir2an 955 . 2 Ord (𝑅1 “ dom card)
52 r1sdom 8634 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
5326, 52sylan 488 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → (𝑅1𝑦) ≺ (𝑅1𝑥))
54 cardsdom2 8811 . . . . . . 7 (((𝑅1𝑦) ∈ dom card ∧ (𝑅1𝑥) ∈ dom card) → ((card‘(𝑅1𝑦)) ∈ (card‘(𝑅1𝑥)) ↔ (𝑅1𝑦) ≺ (𝑅1𝑥)))
5535, 30, 54syl2anc 693 . . . . . 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 6271 . . . . . 6 ((𝑅1 Fn On ∧ 𝑦 ∈ On) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1𝑦)))
584, 28, 57sylancr 695 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑦) = (card‘(𝑅1𝑦)))
5926adantr 481 . . . . . 6 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → 𝑥 ∈ On)
60 fvco2 6271 . . . . . 6 ((𝑅1 Fn On ∧ 𝑥 ∈ On) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1𝑥)))
614, 59, 60sylancr 695 . . . . 5 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑥) = (card‘(𝑅1𝑥)))
6256, 58, 613eltr4d 2715 . . . 4 ((𝑥 ∈ (𝑅1 “ dom card) ∧ 𝑦𝑥) → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥))
6362ex 450 . . 3 (𝑥 ∈ (𝑅1 “ dom card) → (𝑦𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
6463adantl 482 . 2 ((𝑦 ∈ (𝑅1 “ dom card) ∧ 𝑥 ∈ (𝑅1 “ dom card)) → (𝑦𝑥 → ((card ∘ 𝑅1)‘𝑦) ∈ ((card ∘ 𝑅1)‘𝑥)))
6523, 51, 64, 21issmo 7442 1 Smo (card ∘ 𝑅1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  {cab 2607  wral 2911  wrex 2912  Vcvv 3198  wss 3572   class class class wbr 4651  Tr wtr 4750   E cep 5026   We wwe 5070  ccnv 5111  dom cdm 5112  ran crn 5113  cres 5114  cima 5115  ccom 5116  Ord word 5720  Oncon0 5721  Fun wfun 5880   Fn wfn 5881  wf 5882  cfv 5886  Smo wsmo 7439  cen 7949  csdm 7951  𝑅1cr1 8622  cardccrd 8758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-om 7063  df-wrecs 7404  df-smo 7440  df-recs 7465  df-rdg 7503  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-r1 8624  df-card 8762
This theorem is referenced by: (None)
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