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Theorem aomclem4 40494
Description: Lemma for dfac11 40499. Limit case. Patch together well-orderings constructed so far using fnwe2 40490 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 6247 . . 3 (𝑐 = (rank‘𝑎) → suc 𝑐 = suc (rank‘𝑎))
21fveq2d 6690 . 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 9281 . . . . . . . . . . . . . 14 𝑅1 Fn On
5 fnfun 6448 . . . . . . . . . . . . . 14 (𝑅1 Fn On → Fun 𝑅1)
64, 5ax-mp 5 . . . . . . . . . . . . 13 Fun 𝑅1
74fndmi 6451 . . . . . . . . . . . . . 14 dom 𝑅1 = On
87eqimss2i 3946 . . . . . . . . . . . . 13 On ⊆ dom 𝑅1
96, 8pm3.2i 474 . . . . . . . . . . . 12 (Fun 𝑅1 ∧ On ⊆ dom 𝑅1)
10 aomclem4.on . . . . . . . . . . . 12 (𝜑 → dom 𝑧 ∈ On)
11 funfvima2 7016 . . . . . . . . . . . 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 4838 . . . . . . . . . . 11 ((𝑅1‘dom 𝑧) ∈ (𝑅1 “ On) → (𝑅1‘dom 𝑧) ⊆ (𝑅1 “ On))
1412, 13syl 17 . . . . . . . . . 10 (𝜑 → (𝑅1‘dom 𝑧) ⊆ (𝑅1 “ On))
1514sselda 3887 . . . . . . . . 9 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 (𝑅1 “ On))
16 rankidb 9314 . . . . . . . . 9 (𝑏 (𝑅1 “ On) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
1715, 16syl 17 . . . . . . . 8 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
18 suceq 6247 . . . . . . . . . 10 ((rank‘𝑏) = (rank‘𝑎) → suc (rank‘𝑏) = suc (rank‘𝑎))
1918fveq2d 6690 . . . . . . . . 9 ((rank‘𝑏) = (rank‘𝑎) → (𝑅1‘suc (rank‘𝑏)) = (𝑅1‘suc (rank‘𝑎)))
2019eleq2d 2819 . . . . . . . 8 ((rank‘𝑏) = (rank‘𝑎) → (𝑏 ∈ (𝑅1‘suc (rank‘𝑏)) ↔ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2117, 20syl5ibcom 248 . . . . . . 7 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑏) = (rank‘𝑎) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2221expimpd 457 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2322ss2abdv 3963 . . . . 5 (𝜑 → {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))} ⊆ {𝑏𝑏 ∈ (𝑅1‘suc (rank‘𝑎))})
24 df-rab 3063 . . . . 5 {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} = {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))}
25 abid1 2874 . . . . 5 (𝑅1‘suc (rank‘𝑎)) = {𝑏𝑏 ∈ (𝑅1‘suc (rank‘𝑎))}
2623, 24, 253sstr4g 3932 . . . 4 (𝜑 → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
2726adantr 484 . . 3 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
28 fveq2 6686 . . . . 5 (𝑏 = suc (rank‘𝑎) → (𝑧𝑏) = (𝑧‘suc (rank‘𝑎)))
29 fveq2 6686 . . . . 5 (𝑏 = suc (rank‘𝑎) → (𝑅1𝑏) = (𝑅1‘suc (rank‘𝑎)))
3028, 29weeq12d 40477 . . . 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 6686 . . . . . . . 8 (𝑎 = 𝑏 → (𝑧𝑎) = (𝑧𝑏))
33 fveq2 6686 . . . . . . . 8 (𝑎 = 𝑏 → (𝑅1𝑎) = (𝑅1𝑏))
3432, 33weeq12d 40477 . . . . . . 7 (𝑎 = 𝑏 → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝑧𝑏) We (𝑅1𝑏)))
3534cbvralvw 3350 . . . . . 6 (∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎) ↔ ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
3631, 35sylib 221 . . . . 5 (𝜑 → ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
3736adantr 484 . . . 4 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
38 rankr1ai 9312 . . . . . 6 (𝑎 ∈ (𝑅1‘dom 𝑧) → (rank‘𝑎) ∈ dom 𝑧)
3938adantl 485 . . . . 5 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (rank‘𝑎) ∈ dom 𝑧)
40 eloni 6192 . . . . . . . 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 40478 . . . . . . 7 ((Ord dom 𝑧 ∧ dom 𝑧 = dom 𝑧) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
4441, 42, 43syl2anc 587 . . . . . 6 (𝜑 → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
4544adantr 484 . . . . 5 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
4639, 45mpbid 235 . . . 4 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → suc (rank‘𝑎) ∈ dom 𝑧)
4730, 37, 46rspcdva 3531 . . 3 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎)))
48 wess 5522 . . 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 9308 . . . 4 rank: (𝑅1 “ On)⟶On
5150a1i 11 . . 3 (𝜑 → rank: (𝑅1 “ On)⟶On)
5251, 14fssresd 6555 . 2 (𝜑 → (rank ↾ (𝑅1‘dom 𝑧)):(𝑅1‘dom 𝑧)⟶On)
53 epweon 7528 . . 3 E We On
5453a1i 11 . 2 (𝜑 → E We On)
552, 3, 49, 52, 54fnwe2 40490 1 (𝜑𝐹 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 846   = wceq 1542  wcel 2114  {cab 2717  wral 3054  {crab 3058  wss 3853   cuni 4806   class class class wbr 5040  {copab 5102   E cep 5443   We wwe 5492  dom cdm 5535  cima 5538  Ord word 6181  Oncon0 6182  suc csuc 6184  Fun wfun 6343   Fn wfn 6344  wf 6345  cfv 6349  𝑅1cr1 9276  rankcrnk 9277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7612  df-wrecs 7988  df-recs 8049  df-rdg 8087  df-r1 9278  df-rank 9279
This theorem is referenced by:  aomclem5  40495
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