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Theorem aomclem4 40882
Description: Lemma for dfac11 40887. Limit case. Patch together well-orderings constructed so far using fnwe2 40878 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem4.f 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
aomclem4.on (𝜑 → dom 𝑧 ∈ On)
aomclem4.su (𝜑 → dom 𝑧 = dom 𝑧)
aomclem4.we (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
Assertion
Ref Expression
aomclem4 (𝜑𝐹 We (𝑅1‘dom 𝑧))
Distinct variable groups:   𝑧,𝑎,𝑏   𝜑,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑧)   𝐹(𝑧,𝑎,𝑏)

Proof of Theorem aomclem4
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 suceq 6331 . . 3 (𝑐 = (rank‘𝑎) → suc 𝑐 = suc (rank‘𝑎))
21fveq2d 6778 . 2 (𝑐 = (rank‘𝑎) → (𝑧‘suc 𝑐) = (𝑧‘suc (rank‘𝑎)))
3 aomclem4.f . 2 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
4 r1fnon 9525 . . . . . . . . . . . . . 14 𝑅1 Fn On
5 fnfun 6533 . . . . . . . . . . . . . 14 (𝑅1 Fn On → Fun 𝑅1)
64, 5ax-mp 5 . . . . . . . . . . . . 13 Fun 𝑅1
74fndmi 6537 . . . . . . . . . . . . . 14 dom 𝑅1 = On
87eqimss2i 3980 . . . . . . . . . . . . 13 On ⊆ dom 𝑅1
96, 8pm3.2i 471 . . . . . . . . . . . 12 (Fun 𝑅1 ∧ On ⊆ dom 𝑅1)
10 aomclem4.on . . . . . . . . . . . 12 (𝜑 → dom 𝑧 ∈ On)
11 funfvima2 7107 . . . . . . . . . . . 12 ((Fun 𝑅1 ∧ On ⊆ dom 𝑅1) → (dom 𝑧 ∈ On → (𝑅1‘dom 𝑧) ∈ (𝑅1 “ On)))
129, 10, 11mpsyl 68 . . . . . . . . . . 11 (𝜑 → (𝑅1‘dom 𝑧) ∈ (𝑅1 “ On))
13 elssuni 4871 . . . . . . . . . . 11 ((𝑅1‘dom 𝑧) ∈ (𝑅1 “ On) → (𝑅1‘dom 𝑧) ⊆ (𝑅1 “ On))
1412, 13syl 17 . . . . . . . . . 10 (𝜑 → (𝑅1‘dom 𝑧) ⊆ (𝑅1 “ On))
1514sselda 3921 . . . . . . . . 9 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 (𝑅1 “ On))
16 rankidb 9558 . . . . . . . . 9 (𝑏 (𝑅1 “ On) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
1715, 16syl 17 . . . . . . . 8 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
18 suceq 6331 . . . . . . . . . 10 ((rank‘𝑏) = (rank‘𝑎) → suc (rank‘𝑏) = suc (rank‘𝑎))
1918fveq2d 6778 . . . . . . . . 9 ((rank‘𝑏) = (rank‘𝑎) → (𝑅1‘suc (rank‘𝑏)) = (𝑅1‘suc (rank‘𝑎)))
2019eleq2d 2824 . . . . . . . 8 ((rank‘𝑏) = (rank‘𝑎) → (𝑏 ∈ (𝑅1‘suc (rank‘𝑏)) ↔ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2117, 20syl5ibcom 244 . . . . . . 7 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑏) = (rank‘𝑎) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2221expimpd 454 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2322ss2abdv 3997 . . . . 5 (𝜑 → {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))} ⊆ {𝑏𝑏 ∈ (𝑅1‘suc (rank‘𝑎))})
24 df-rab 3073 . . . . 5 {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} = {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))}
25 abid1 2881 . . . . 5 (𝑅1‘suc (rank‘𝑎)) = {𝑏𝑏 ∈ (𝑅1‘suc (rank‘𝑎))}
2623, 24, 253sstr4g 3966 . . . 4 (𝜑 → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
2726adantr 481 . . 3 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
28 fveq2 6774 . . . . 5 (𝑏 = suc (rank‘𝑎) → (𝑧𝑏) = (𝑧‘suc (rank‘𝑎)))
29 fveq2 6774 . . . . 5 (𝑏 = suc (rank‘𝑎) → (𝑅1𝑏) = (𝑅1‘suc (rank‘𝑎)))
3028, 29weeq12d 40865 . . . 4 (𝑏 = suc (rank‘𝑎) → ((𝑧𝑏) We (𝑅1𝑏) ↔ (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎))))
31 aomclem4.we . . . . . 6 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
32 fveq2 6774 . . . . . . . 8 (𝑎 = 𝑏 → (𝑧𝑎) = (𝑧𝑏))
33 fveq2 6774 . . . . . . . 8 (𝑎 = 𝑏 → (𝑅1𝑎) = (𝑅1𝑏))
3432, 33weeq12d 40865 . . . . . . 7 (𝑎 = 𝑏 → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝑧𝑏) We (𝑅1𝑏)))
3534cbvralvw 3383 . . . . . 6 (∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎) ↔ ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
3631, 35sylib 217 . . . . 5 (𝜑 → ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
3736adantr 481 . . . 4 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
38 rankr1ai 9556 . . . . . 6 (𝑎 ∈ (𝑅1‘dom 𝑧) → (rank‘𝑎) ∈ dom 𝑧)
3938adantl 482 . . . . 5 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (rank‘𝑎) ∈ dom 𝑧)
40 eloni 6276 . . . . . . . 8 (dom 𝑧 ∈ On → Ord dom 𝑧)
4110, 40syl 17 . . . . . . 7 (𝜑 → Ord dom 𝑧)
42 aomclem4.su . . . . . . 7 (𝜑 → dom 𝑧 = dom 𝑧)
43 limsuc2 40866 . . . . . . 7 ((Ord dom 𝑧 ∧ dom 𝑧 = dom 𝑧) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
4441, 42, 43syl2anc 584 . . . . . 6 (𝜑 → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
4544adantr 481 . . . . 5 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
4639, 45mpbid 231 . . . 4 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → suc (rank‘𝑎) ∈ dom 𝑧)
4730, 37, 46rspcdva 3562 . . 3 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎)))
48 wess 5576 . . 3 ({𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)) → ((𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎)) → (𝑧‘suc (rank‘𝑎)) We {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)}))
4927, 47, 48sylc 65 . 2 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)})
50 rankf 9552 . . . 4 rank: (𝑅1 “ On)⟶On
5150a1i 11 . . 3 (𝜑 → rank: (𝑅1 “ On)⟶On)
5251, 14fssresd 6641 . 2 (𝜑 → (rank ↾ (𝑅1‘dom 𝑧)):(𝑅1‘dom 𝑧)⟶On)
53 epweon 7625 . . 3 E We On
5453a1i 11 . 2 (𝜑 → E We On)
552, 3, 49, 52, 54fnwe2 40878 1 (𝜑𝐹 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844   = wceq 1539  wcel 2106  {cab 2715  wral 3064  {crab 3068  wss 3887   cuni 4839   class class class wbr 5074  {copab 5136   E cep 5494   We wwe 5543  dom cdm 5589  cima 5592  Ord word 6265  Oncon0 6266  suc csuc 6268  Fun wfun 6427   Fn wfn 6428  wf 6429  cfv 6433  𝑅1cr1 9520  rankcrnk 9521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-rank 9523
This theorem is referenced by:  aomclem5  40883
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