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Theorem aomclem4 38236
Description: Lemma for dfac11 38241. Limit case. Patch together well-orderings constructed so far using fnwe2 38232 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 5973 . . 3 (𝑐 = (rank‘𝑎) → suc 𝑐 = suc (rank‘𝑎))
21fveq2d 6379 . 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 8845 . . . . . . . . . . . . . 14 𝑅1 Fn On
5 fnfun 6166 . . . . . . . . . . . . . 14 (𝑅1 Fn On → Fun 𝑅1)
64, 5ax-mp 5 . . . . . . . . . . . . 13 Fun 𝑅1
7 fndm 6168 . . . . . . . . . . . . . . 15 (𝑅1 Fn On → dom 𝑅1 = On)
84, 7ax-mp 5 . . . . . . . . . . . . . 14 dom 𝑅1 = On
98eqimss2i 3820 . . . . . . . . . . . . 13 On ⊆ dom 𝑅1
106, 9pm3.2i 462 . . . . . . . . . . . 12 (Fun 𝑅1 ∧ On ⊆ dom 𝑅1)
11 aomclem4.on . . . . . . . . . . . 12 (𝜑 → dom 𝑧 ∈ On)
12 funfvima2 6686 . . . . . . . . . . . 12 ((Fun 𝑅1 ∧ On ⊆ dom 𝑅1) → (dom 𝑧 ∈ On → (𝑅1‘dom 𝑧) ∈ (𝑅1 “ On)))
1310, 11, 12mpsyl 68 . . . . . . . . . . 11 (𝜑 → (𝑅1‘dom 𝑧) ∈ (𝑅1 “ On))
14 elssuni 4625 . . . . . . . . . . 11 ((𝑅1‘dom 𝑧) ∈ (𝑅1 “ On) → (𝑅1‘dom 𝑧) ⊆ (𝑅1 “ On))
1513, 14syl 17 . . . . . . . . . 10 (𝜑 → (𝑅1‘dom 𝑧) ⊆ (𝑅1 “ On))
1615sselda 3761 . . . . . . . . 9 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 (𝑅1 “ On))
17 rankidb 8878 . . . . . . . . 9 (𝑏 (𝑅1 “ On) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
1816, 17syl 17 . . . . . . . 8 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
19 suceq 5973 . . . . . . . . . 10 ((rank‘𝑏) = (rank‘𝑎) → suc (rank‘𝑏) = suc (rank‘𝑎))
2019fveq2d 6379 . . . . . . . . 9 ((rank‘𝑏) = (rank‘𝑎) → (𝑅1‘suc (rank‘𝑏)) = (𝑅1‘suc (rank‘𝑎)))
2120eleq2d 2830 . . . . . . . 8 ((rank‘𝑏) = (rank‘𝑎) → (𝑏 ∈ (𝑅1‘suc (rank‘𝑏)) ↔ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2218, 21syl5ibcom 236 . . . . . . 7 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑏) = (rank‘𝑎) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2322expimpd 445 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2423ss2abdv 3835 . . . . 5 (𝜑 → {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))} ⊆ {𝑏𝑏 ∈ (𝑅1‘suc (rank‘𝑎))})
25 df-rab 3064 . . . . 5 {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} = {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))}
26 abid1 2887 . . . . 5 (𝑅1‘suc (rank‘𝑎)) = {𝑏𝑏 ∈ (𝑅1‘suc (rank‘𝑎))}
2724, 25, 263sstr4g 3806 . . . 4 (𝜑 → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
2827adantr 472 . . 3 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
29 rankr1ai 8876 . . . . . 6 (𝑎 ∈ (𝑅1‘dom 𝑧) → (rank‘𝑎) ∈ dom 𝑧)
3029adantl 473 . . . . 5 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (rank‘𝑎) ∈ dom 𝑧)
31 eloni 5918 . . . . . . . 8 (dom 𝑧 ∈ On → Ord dom 𝑧)
3211, 31syl 17 . . . . . . 7 (𝜑 → Ord dom 𝑧)
33 aomclem4.su . . . . . . 7 (𝜑 → dom 𝑧 = dom 𝑧)
34 limsuc2 38220 . . . . . . 7 ((Ord dom 𝑧 ∧ dom 𝑧 = dom 𝑧) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
3532, 33, 34syl2anc 579 . . . . . 6 (𝜑 → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
3635adantr 472 . . . . 5 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
3730, 36mpbid 223 . . . 4 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → suc (rank‘𝑎) ∈ dom 𝑧)
38 aomclem4.we . . . . . 6 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
39 fveq2 6375 . . . . . . . 8 (𝑎 = 𝑏 → (𝑧𝑎) = (𝑧𝑏))
40 fveq2 6375 . . . . . . . 8 (𝑎 = 𝑏 → (𝑅1𝑎) = (𝑅1𝑏))
4139, 40weeq12d 38219 . . . . . . 7 (𝑎 = 𝑏 → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝑧𝑏) We (𝑅1𝑏)))
4241cbvralv 3319 . . . . . 6 (∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎) ↔ ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
4338, 42sylib 209 . . . . 5 (𝜑 → ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
4443adantr 472 . . . 4 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
45 fveq2 6375 . . . . . 6 (𝑏 = suc (rank‘𝑎) → (𝑧𝑏) = (𝑧‘suc (rank‘𝑎)))
46 fveq2 6375 . . . . . 6 (𝑏 = suc (rank‘𝑎) → (𝑅1𝑏) = (𝑅1‘suc (rank‘𝑎)))
4745, 46weeq12d 38219 . . . . 5 (𝑏 = suc (rank‘𝑎) → ((𝑧𝑏) We (𝑅1𝑏) ↔ (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎))))
4847rspcva 3459 . . . 4 ((suc (rank‘𝑎) ∈ dom 𝑧 ∧ ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏)) → (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎)))
4937, 44, 48syl2anc 579 . . 3 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎)))
50 wess 5264 . . 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)}))
5128, 49, 50sylc 65 . 2 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)})
52 rankf 8872 . . . 4 rank: (𝑅1 “ On)⟶On
5352a1i 11 . . 3 (𝜑 → rank: (𝑅1 “ On)⟶On)
5453, 15fssresd 6253 . 2 (𝜑 → (rank ↾ (𝑅1‘dom 𝑧)):(𝑅1‘dom 𝑧)⟶On)
55 epweon 7181 . . 3 E We On
5655a1i 11 . 2 (𝜑 → E We On)
572, 3, 51, 54, 56fnwe2 38232 1 (𝜑𝐹 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  {cab 2751  wral 3055  {crab 3059  wss 3732   cuni 4594   class class class wbr 4809  {copab 4871   E cep 5189   We wwe 5235  dom cdm 5277  cima 5280  Ord word 5907  Oncon0 5908  suc csuc 5910  Fun wfun 6062   Fn wfn 6063  wf 6064  cfv 6068  𝑅1cr1 8840  rankcrnk 8841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-r1 8842  df-rank 8843
This theorem is referenced by:  aomclem5  38237
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