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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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