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Theorem aomclem4 38147
Description: Lemma for dfac11 38152. Limit case. Patch together well-orderings constructed so far using fnwe2 38143 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 5951 . . 3 (𝑐 = (rank‘𝑎) → suc 𝑐 = suc (rank‘𝑎))
21fveq2d 6357 . 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 8805 . . . . . . . . . . . . . 14 𝑅1 Fn On
5 fnfun 6149 . . . . . . . . . . . . . 14 (𝑅1 Fn On → Fun 𝑅1)
64, 5ax-mp 5 . . . . . . . . . . . . 13 Fun 𝑅1
7 fndm 6151 . . . . . . . . . . . . . . 15 (𝑅1 Fn On → dom 𝑅1 = On)
84, 7ax-mp 5 . . . . . . . . . . . . . 14 dom 𝑅1 = On
98eqimss2i 3801 . . . . . . . . . . . . 13 On ⊆ dom 𝑅1
106, 9pm3.2i 470 . . . . . . . . . . . 12 (Fun 𝑅1 ∧ On ⊆ dom 𝑅1)
11 aomclem4.on . . . . . . . . . . . 12 (𝜑 → dom 𝑧 ∈ On)
12 funfvima2 6657 . . . . . . . . . . . 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 4619 . . . . . . . . . . 11 ((𝑅1‘dom 𝑧) ∈ (𝑅1 “ On) → (𝑅1‘dom 𝑧) ⊆ (𝑅1 “ On))
1513, 14syl 17 . . . . . . . . . 10 (𝜑 → (𝑅1‘dom 𝑧) ⊆ (𝑅1 “ On))
1615sselda 3744 . . . . . . . . 9 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 (𝑅1 “ On))
17 rankidb 8838 . . . . . . . . 9 (𝑏 (𝑅1 “ On) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
1816, 17syl 17 . . . . . . . 8 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
19 suceq 5951 . . . . . . . . . 10 ((rank‘𝑏) = (rank‘𝑎) → suc (rank‘𝑏) = suc (rank‘𝑎))
2019fveq2d 6357 . . . . . . . . 9 ((rank‘𝑏) = (rank‘𝑎) → (𝑅1‘suc (rank‘𝑏)) = (𝑅1‘suc (rank‘𝑎)))
2120eleq2d 2825 . . . . . . . 8 ((rank‘𝑏) = (rank‘𝑎) → (𝑏 ∈ (𝑅1‘suc (rank‘𝑏)) ↔ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2218, 21syl5ibcom 235 . . . . . . 7 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑏) = (rank‘𝑎) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2322expimpd 630 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2423ss2abdv 3816 . . . . 5 (𝜑 → {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))} ⊆ {𝑏𝑏 ∈ (𝑅1‘suc (rank‘𝑎))})
25 df-rab 3059 . . . . 5 {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} = {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))}
26 abid1 2882 . . . . 5 (𝑅1‘suc (rank‘𝑎)) = {𝑏𝑏 ∈ (𝑅1‘suc (rank‘𝑎))}
2724, 25, 263sstr4g 3787 . . . 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 8836 . . . . . 6 (𝑎 ∈ (𝑅1‘dom 𝑧) → (rank‘𝑎) ∈ dom 𝑧)
3029adantl 473 . . . . 5 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (rank‘𝑎) ∈ dom 𝑧)
31 eloni 5894 . . . . . . . 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 38131 . . . . . . 7 ((Ord dom 𝑧 ∧ dom 𝑧 = dom 𝑧) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
3532, 33, 34syl2anc 696 . . . . . 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 222 . . . 4 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → suc (rank‘𝑎) ∈ dom 𝑧)
38 aomclem4.we . . . . . 6 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
39 fveq2 6353 . . . . . . . 8 (𝑎 = 𝑏 → (𝑧𝑎) = (𝑧𝑏))
40 fveq2 6353 . . . . . . . 8 (𝑎 = 𝑏 → (𝑅1𝑎) = (𝑅1𝑏))
4139, 40weeq12d 38130 . . . . . . 7 (𝑎 = 𝑏 → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝑧𝑏) We (𝑅1𝑏)))
4241cbvralv 3310 . . . . . 6 (∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎) ↔ ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
4338, 42sylib 208 . . . . 5 (𝜑 → ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
4443adantr 472 . . . 4 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
45 fveq2 6353 . . . . . 6 (𝑏 = suc (rank‘𝑎) → (𝑧𝑏) = (𝑧‘suc (rank‘𝑎)))
46 fveq2 6353 . . . . . 6 (𝑏 = suc (rank‘𝑎) → (𝑅1𝑏) = (𝑅1‘suc (rank‘𝑎)))
4745, 46weeq12d 38130 . . . . 5 (𝑏 = suc (rank‘𝑎) → ((𝑧𝑏) We (𝑅1𝑏) ↔ (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎))))
4847rspcva 3447 . . . 4 ((suc (rank‘𝑎) ∈ dom 𝑧 ∧ ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏)) → (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎)))
4937, 44, 48syl2anc 696 . . 3 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎)))
50 wess 5253 . . 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 8832 . . . 4 rank: (𝑅1 “ On)⟶On
5352a1i 11 . . 3 (𝜑 → rank: (𝑅1 “ On)⟶On)
5453, 15fssresd 6232 . 2 (𝜑 → (rank ↾ (𝑅1‘dom 𝑧)):(𝑅1‘dom 𝑧)⟶On)
55 epweon 7149 . . 3 E We On
5655a1i 11 . 2 (𝜑 → E We On)
572, 3, 51, 54, 56fnwe2 38143 1 (𝜑𝐹 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  {crab 3054  wss 3715   cuni 4588   class class class wbr 4804  {copab 4864   E cep 5178   We wwe 5224  dom cdm 5266  cima 5269  Ord word 5883  Oncon0 5884  suc csuc 5886  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  𝑅1cr1 8800  rankcrnk 8801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-r1 8802  df-rank 8803
This theorem is referenced by:  aomclem5  38148
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