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Theorem rankr1ai 9203
Description: One direction of rankr1a 9241. (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 6675 . . 3 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ dom 𝑅1)
2 r1val1 9191 . . . . . 6 (𝐵 ∈ dom 𝑅1 → (𝑅1𝐵) = 𝑥𝐵 𝒫 (𝑅1𝑥))
32eleq2d 2897 . . . . 5 (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1𝐵) ↔ 𝐴 𝑥𝐵 𝒫 (𝑅1𝑥)))
4 eliun 4896 . . . . 5 (𝐴 𝑥𝐵 𝒫 (𝑅1𝑥) ↔ ∃𝑥𝐵 𝐴 ∈ 𝒫 (𝑅1𝑥))
53, 4syl6bb 290 . . . 4 (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1𝐵) ↔ ∃𝑥𝐵 𝐴 ∈ 𝒫 (𝑅1𝑥)))
6 r1funlim 9171 . . . . . . . . . . 11 (Fun 𝑅1 ∧ Lim dom 𝑅1)
76simpri 489 . . . . . . . . . 10 Lim dom 𝑅1
8 limord 6223 . . . . . . . . . 10 (Lim dom 𝑅1 → Ord dom 𝑅1)
97, 8ax-mp 5 . . . . . . . . 9 Ord dom 𝑅1
10 ordtr1 6207 . . . . . . . . 9 (Ord dom 𝑅1 → ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
119, 10ax-mp 5 . . . . . . . 8 ((𝑥𝐵𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
1211ancoms 462 . . . . . . 7 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → 𝑥 ∈ dom 𝑅1)
13 r1sucg 9174 . . . . . . . 8 (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
1413eleq2d 2897 . . . . . . 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 7479 . . . . . . . . . 10 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
179, 16ax-mp 5 . . . . . . . . 9 dom 𝑅1 ⊆ On
1817, 12sseldi 3941 . . . . . . . 8 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → 𝑥 ∈ On)
19 rabid 3363 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ↔ (𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)))
20 intss1 4864 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
2119, 20sylbir 238 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
2218, 21sylan 583 . . . . . . 7 (((𝐵 ∈ dom 𝑅1𝑥𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
2322ex 416 . . . . . 6 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → (𝐴 ∈ (𝑅1‘suc 𝑥) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
2415, 23sylbird 263 . . . . 5 ((𝐵 ∈ dom 𝑅1𝑥𝐵) → (𝐴 ∈ 𝒫 (𝑅1𝑥) → {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
2524reximdva 3260 . . . 4 (𝐵 ∈ dom 𝑅1 → (∃𝑥𝐵 𝐴 ∈ 𝒫 (𝑅1𝑥) → ∃𝑥𝐵 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
265, 25sylbid 243 . . 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 9201 . . . . . . 7 (𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
29 rankvalb 9202 . . . . . . 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 3974 . . . . 5 (𝐴 ∈ (𝑅1𝐵) → ((rank‘𝐴) ⊆ 𝑥 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
3231adantr 484 . . . 4 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝑥𝐵) → ((rank‘𝐴) ⊆ 𝑥 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
33 rankon 9200 . . . . . . 7 (rank‘𝐴) ∈ On
3417, 1sseldi 3941 . . . . . . 7 (𝐴 ∈ (𝑅1𝐵) → 𝐵 ∈ On)
35 ontr2 6211 . . . . . . 7 (((rank‘𝐴) ∈ On ∧ 𝐵 ∈ On) → (((rank‘𝐴) ⊆ 𝑥𝑥𝐵) → (rank‘𝐴) ∈ 𝐵))
3633, 34, 35sylancr 590 . . . . . 6 (𝐴 ∈ (𝑅1𝐵) → (((rank‘𝐴) ⊆ 𝑥𝑥𝐵) → (rank‘𝐴) ∈ 𝐵))
3736expcomd 420 . . . . 5 (𝐴 ∈ (𝑅1𝐵) → (𝑥𝐵 → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵)))
3837imp 410 . . . 4 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝑥𝐵) → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
3932, 38sylbird 263 . . 3 ((𝐴 ∈ (𝑅1𝐵) ∧ 𝑥𝐵) → ( {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
4039rexlimdva 3270 . 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 209  wa 399   = wceq 1538  wcel 2115  wrex 3127  {crab 3130  wss 3910  𝒫 cpw 4512   cuni 4811   cint 4849   ciun 4892  dom cdm 5528  cima 5531  Ord word 6163  Oncon0 6164  Lim wlim 6165  suc csuc 6166  Fun wfun 6322  cfv 6328  𝑅1cr1 9167  rankcrnk 9168
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 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-r1 9169  df-rank 9170
This theorem is referenced by:  rankr1ag  9207  tcrank  9289  dfac12lem1  9546  dfac12lem2  9547  r1limwun  10135  inatsk  10177  aomclem4  39796  r1rankcld  40722
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