MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankr1ai Structured version   Visualization version   GIF version

Theorem rankr1ai 9835
Description: One direction of rankr1a 9873. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankr1ai (𝐴 ∈ (𝑅1𝐵) → (rank‘𝐴) ∈ 𝐵)

Proof of Theorem rankr1ai
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6943 . . 3 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
2 r1val1 9823 . . . . . 6 (𝐵 ∈ dom 𝑅1 → (𝑅1𝐵) = 𝑥𝐵 𝒫 (𝑅1𝑥))
32eleq2d 2824 . . . . 5 (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1𝐵) ↔ 𝐴 𝑥𝐵 𝒫 (𝑅1𝑥)))
4 eliun 4999 . . . . 5 (𝐴 𝑥𝐵 𝒫 (𝑅1𝑥) ↔ ∃𝑥𝐵 𝐴 ∈ 𝒫 (𝑅1𝑥))
53, 4bitrdi 287 . . . 4 (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1𝐵) ↔ ∃𝑥𝐵 𝐴 ∈ 𝒫 (𝑅1𝑥)))
6 r1funlim 9803 . . . . . . . . . . 11 (Fun 𝑅1 ∧ Lim dom 𝑅1)
76simpri 485 . . . . . . . . . 10 Lim dom 𝑅1
8 limord 6445 . . . . . . . . . 10 (Lim dom 𝑅1 → Ord dom 𝑅1)
97, 8ax-mp 5 . . . . . . . . 9 Ord dom 𝑅1
10 ordtr1 6428 . . . . . . . . 9 (Ord dom 𝑅1 → ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
119, 10ax-mp 5 . . . . . . . 8 ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
1211ancoms 458 . . . . . . 7 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → 𝑥 ∈ dom 𝑅1)
13 r1sucg 9806 . . . . . . . 8 (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
1413eleq2d 2824 . . . . . . 7 (𝑥 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ 𝒫 (𝑅1𝑥)))
1512, 14syl 17 . . . . . 6 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ 𝒫 (𝑅1𝑥)))
16 ordsson 7801 . . . . . . . . . 10 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
179, 16ax-mp 5 . . . . . . . . 9 dom 𝑅1 ⊆ On
1817, 12sselid 3992 . . . . . . . 8 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → 𝑥 ∈ On)
19 rabid 3454 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ↔ (𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)))
20 intss1 4967 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
2119, 20sylbir 235 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
2218, 21sylan 580 . . . . . . 7 (((𝐵 ∈ dom 𝑅1𝑥𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
2322ex 412 . . . . . 6 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → (𝐴 ∈ (𝑅1‘suc 𝑥) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
2415, 23sylbird 260 . . . . 5 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → (𝐴 ∈ 𝒫 (𝑅1𝑥) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
2524reximdva 3165 . . . 4 (𝐵 ∈ dom 𝑅1 → (∃𝑥𝐵 𝐴 ∈ 𝒫 (𝑅1𝑥) → ∃𝑥𝐵 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
265, 25sylbid 240 . . 3 (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1𝐵) → ∃𝑥𝐵 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
271, 26mpcom 38 . 2 (𝐴 ∈ (𝑅1𝐵) → ∃𝑥𝐵 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
28 r1elwf 9833 . . . . . . 7 (𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
29 rankvalb 9834 . . . . . . 7 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
3028, 29syl 17 . . . . . 6 (𝐴 ∈ (𝑅1𝐵) → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
3130sseq1d 4026 . . . . 5 (𝐴 ∈ (𝑅1𝐵) → ((rank‘𝐴) ⊆ 𝑥 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
3231adantr 480 . . . 4 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝑥𝐵) → ((rank‘𝐴) ⊆ 𝑥 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
33 rankon 9832 . . . . . . 7 (rank‘𝐴) ∈ On
3417, 1sselid 3992 . . . . . . 7 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ On)
35 ontr2 6432 . . . . . . 7 (((rank‘𝐴) ∈ On ∧ 𝐵 ∈ On) → (((rank‘𝐴) ⊆ 𝑥𝑥𝐵) → (rank‘𝐴) ∈ 𝐵))
3633, 34, 35sylancr 587 . . . . . 6 (𝐴 ∈ (𝑅1𝐵) → (((rank‘𝐴) ⊆ 𝑥𝑥𝐵) → (rank‘𝐴) ∈ 𝐵))
3736expcomd 416 . . . . 5 (𝐴 ∈ (𝑅1𝐵) → (𝑥𝐵 → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵)))
3837imp 406 . . . 4 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝑥𝐵) → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
3932, 38sylbird 260 . . 3 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝑥𝐵) → ( {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
4039rexlimdva 3152 . 2 (𝐴 ∈ (𝑅1𝐵) → (∃𝑥𝐵 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
4127, 40mpd 15 1 (𝐴 ∈ (𝑅1𝐵) → (rank‘𝐴) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wrex 3067  {crab 3432  wss 3962  𝒫 cpw 4604   cuni 4911   cint 4950   ciun 4995  dom cdm 5688  cima 5691  Ord word 6384  Oncon0 6385  Lim wlim 6386  suc csuc 6387  Fun wfun 6556  cfv 6562  𝑅1cr1 9799  rankcrnk 9800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-r1 9801  df-rank 9802
This theorem is referenced by:  rankr1ag  9839  tcrank  9921  dfac12lem1  10181  dfac12lem2  10182  r1limwun  10773  inatsk  10815  aomclem4  43045  r1rankcld  44226
  Copyright terms: Public domain W3C validator