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Theorem rankr1ai 8613
Description: One direction of rankr1a 8651. (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 6182 . . 3 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
2 r1val1 8601 . . . . . 6 (𝐵 ∈ dom 𝑅1 → (𝑅1𝐵) = 𝑥𝐵 𝒫 (𝑅1𝑥))
32eleq2d 2684 . . . . 5 (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1𝐵) ↔ 𝐴 𝑥𝐵 𝒫 (𝑅1𝑥)))
4 eliun 4495 . . . . 5 (𝐴 𝑥𝐵 𝒫 (𝑅1𝑥) ↔ ∃𝑥𝐵 𝐴 ∈ 𝒫 (𝑅1𝑥))
53, 4syl6bb 276 . . . 4 (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1𝐵) ↔ ∃𝑥𝐵 𝐴 ∈ 𝒫 (𝑅1𝑥)))
6 r1funlim 8581 . . . . . . . . . . 11 (Fun 𝑅1 ∧ Lim dom 𝑅1)
76simpri 478 . . . . . . . . . 10 Lim dom 𝑅1
8 limord 5748 . . . . . . . . . 10 (Lim dom 𝑅1 → Ord dom 𝑅1)
97, 8ax-mp 5 . . . . . . . . 9 Ord dom 𝑅1
10 ordtr1 5731 . . . . . . . . 9 (Ord dom 𝑅1 → ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
119, 10ax-mp 5 . . . . . . . 8 ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
1211ancoms 469 . . . . . . 7 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → 𝑥 ∈ dom 𝑅1)
13 r1sucg 8584 . . . . . . . 8 (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
1413eleq2d 2684 . . . . . . 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 6943 . . . . . . . . . 10 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
179, 16ax-mp 5 . . . . . . . . 9 dom 𝑅1 ⊆ On
1817, 12sseldi 3585 . . . . . . . 8 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → 𝑥 ∈ On)
19 rabid 3109 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ↔ (𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)))
20 intss1 4462 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
2119, 20sylbir 225 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
2218, 21sylan 488 . . . . . . 7 (((𝐵 ∈ dom 𝑅1𝑥𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
2322ex 450 . . . . . 6 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → (𝐴 ∈ (𝑅1‘suc 𝑥) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
2415, 23sylbird 250 . . . . 5 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → (𝐴 ∈ 𝒫 (𝑅1𝑥) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
2524reximdva 3012 . . . 4 (𝐵 ∈ dom 𝑅1 → (∃𝑥𝐵 𝐴 ∈ 𝒫 (𝑅1𝑥) → ∃𝑥𝐵 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
265, 25sylbid 230 . . 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 8611 . . . . . . 7 (𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
29 rankvalb 8612 . . . . . . 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 3616 . . . . 5 (𝐴 ∈ (𝑅1𝐵) → ((rank‘𝐴) ⊆ 𝑥 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
3231adantr 481 . . . 4 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝑥𝐵) → ((rank‘𝐴) ⊆ 𝑥 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
33 rankon 8610 . . . . . . 7 (rank‘𝐴) ∈ On
3417, 1sseldi 3585 . . . . . . 7 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ On)
35 ontr2 5736 . . . . . . 7 (((rank‘𝐴) ∈ On ∧ 𝐵 ∈ On) → (((rank‘𝐴) ⊆ 𝑥𝑥𝐵) → (rank‘𝐴) ∈ 𝐵))
3633, 34, 35sylancr 694 . . . . . 6 (𝐴 ∈ (𝑅1𝐵) → (((rank‘𝐴) ⊆ 𝑥𝑥𝐵) → (rank‘𝐴) ∈ 𝐵))
3736expcomd 454 . . . . 5 (𝐴 ∈ (𝑅1𝐵) → (𝑥𝐵 → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵)))
3837imp 445 . . . 4 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝑥𝐵) → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
3932, 38sylbird 250 . . 3 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝑥𝐵) → ( {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
4039rexlimdva 3025 . 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 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  {crab 2911  wss 3559  𝒫 cpw 4135   cuni 4407   cint 4445   ciun 4490  dom cdm 5079  cima 5082  Ord word 5686  Oncon0 5687  Lim wlim 5688  suc csuc 5689  Fun wfun 5846  cfv 5852  𝑅1cr1 8577  rankcrnk 8578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-r1 8579  df-rank 8580
This theorem is referenced by:  rankr1ag  8617  tcrank  8699  dfac12lem1  8917  dfac12lem2  8918  r1limwun  9510  inatsk  9552  aomclem4  37142
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