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Theorem aomclem4 39541
Description: Lemma for dfac11 39546. Limit case. Patch together well-orderings constructed so far using fnwe2 39537 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 6255 . . 3 (𝑐 = (rank‘𝑎) → suc 𝑐 = suc (rank‘𝑎))
21fveq2d 6673 . 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 9190 . . . . . . . . . . . . . 14 𝑅1 Fn On
5 fnfun 6452 . . . . . . . . . . . . . 14 (𝑅1 Fn On → Fun 𝑅1)
64, 5ax-mp 5 . . . . . . . . . . . . 13 Fun 𝑅1
7 fndm 6454 . . . . . . . . . . . . . . 15 (𝑅1 Fn On → dom 𝑅1 = On)
84, 7ax-mp 5 . . . . . . . . . . . . . 14 dom 𝑅1 = On
98eqimss2i 4030 . . . . . . . . . . . . 13 On ⊆ dom 𝑅1
106, 9pm3.2i 471 . . . . . . . . . . . 12 (Fun 𝑅1 ∧ On ⊆ dom 𝑅1)
11 aomclem4.on . . . . . . . . . . . 12 (𝜑 → dom 𝑧 ∈ On)
12 funfvima2 6990 . . . . . . . . . . . 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 4866 . . . . . . . . . . 11 ((𝑅1‘dom 𝑧) ∈ (𝑅1 “ On) → (𝑅1‘dom 𝑧) ⊆ (𝑅1 “ On))
1513, 14syl 17 . . . . . . . . . 10 (𝜑 → (𝑅1‘dom 𝑧) ⊆ (𝑅1 “ On))
1615sselda 3971 . . . . . . . . 9 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 (𝑅1 “ On))
17 rankidb 9223 . . . . . . . . 9 (𝑏 (𝑅1 “ On) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
1816, 17syl 17 . . . . . . . 8 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
19 suceq 6255 . . . . . . . . . 10 ((rank‘𝑏) = (rank‘𝑎) → suc (rank‘𝑏) = suc (rank‘𝑎))
2019fveq2d 6673 . . . . . . . . 9 ((rank‘𝑏) = (rank‘𝑎) → (𝑅1‘suc (rank‘𝑏)) = (𝑅1‘suc (rank‘𝑎)))
2120eleq2d 2903 . . . . . . . 8 ((rank‘𝑏) = (rank‘𝑎) → (𝑏 ∈ (𝑅1‘suc (rank‘𝑏)) ↔ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2218, 21syl5ibcom 246 . . . . . . 7 ((𝜑𝑏 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑏) = (rank‘𝑎) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2322expimpd 454 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
2423ss2abdv 4048 . . . . 5 (𝜑 → {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))} ⊆ {𝑏𝑏 ∈ (𝑅1‘suc (rank‘𝑎))})
25 df-rab 3152 . . . . 5 {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} = {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))}
26 abid1 2961 . . . . 5 (𝑅1‘suc (rank‘𝑎)) = {𝑏𝑏 ∈ (𝑅1‘suc (rank‘𝑎))}
2724, 25, 263sstr4g 4016 . . . 4 (𝜑 → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
2827adantr 481 . . 3 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
29 fveq2 6669 . . . . 5 (𝑏 = suc (rank‘𝑎) → (𝑧𝑏) = (𝑧‘suc (rank‘𝑎)))
30 fveq2 6669 . . . . 5 (𝑏 = suc (rank‘𝑎) → (𝑅1𝑏) = (𝑅1‘suc (rank‘𝑎)))
3129, 30weeq12d 39524 . . . 4 (𝑏 = suc (rank‘𝑎) → ((𝑧𝑏) We (𝑅1𝑏) ↔ (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎))))
32 aomclem4.we . . . . . 6 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
33 fveq2 6669 . . . . . . . 8 (𝑎 = 𝑏 → (𝑧𝑎) = (𝑧𝑏))
34 fveq2 6669 . . . . . . . 8 (𝑎 = 𝑏 → (𝑅1𝑎) = (𝑅1𝑏))
3533, 34weeq12d 39524 . . . . . . 7 (𝑎 = 𝑏 → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝑧𝑏) We (𝑅1𝑏)))
3635cbvralv 3458 . . . . . 6 (∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎) ↔ ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
3732, 36sylib 219 . . . . 5 (𝜑 → ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
3837adantr 481 . . . 4 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → ∀𝑏 ∈ dom 𝑧(𝑧𝑏) We (𝑅1𝑏))
39 rankr1ai 9221 . . . . . 6 (𝑎 ∈ (𝑅1‘dom 𝑧) → (rank‘𝑎) ∈ dom 𝑧)
4039adantl 482 . . . . 5 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (rank‘𝑎) ∈ dom 𝑧)
41 eloni 6200 . . . . . . . 8 (dom 𝑧 ∈ On → Ord dom 𝑧)
4211, 41syl 17 . . . . . . 7 (𝜑 → Ord dom 𝑧)
43 aomclem4.su . . . . . . 7 (𝜑 → dom 𝑧 = dom 𝑧)
44 limsuc2 39525 . . . . . . 7 ((Ord dom 𝑧 ∧ dom 𝑧 = dom 𝑧) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
4542, 43, 44syl2anc 584 . . . . . 6 (𝜑 → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
4645adantr 481 . . . . 5 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
4740, 46mpbid 233 . . . 4 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → suc (rank‘𝑎) ∈ dom 𝑧)
4831, 38, 47rspcdva 3629 . . 3 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎)))
49 wess 5541 . . 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)}))
5028, 48, 49sylc 65 . 2 ((𝜑𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)})
51 rankf 9217 . . . 4 rank: (𝑅1 “ On)⟶On
5251a1i 11 . . 3 (𝜑 → rank: (𝑅1 “ On)⟶On)
5352, 15fssresd 6544 . 2 (𝜑 → (rank ↾ (𝑅1‘dom 𝑧)):(𝑅1‘dom 𝑧)⟶On)
54 epweon 7490 . . 3 E We On
5554a1i 11 . 2 (𝜑 → E We On)
562, 3, 50, 53, 55fnwe2 39537 1 (𝜑𝐹 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wcel 2107  {cab 2804  wral 3143  {crab 3147  wss 3940   cuni 4837   class class class wbr 5063  {copab 5125   E cep 5463   We wwe 5512  dom cdm 5554  cima 5557  Ord word 6189  Oncon0 6190  suc csuc 6192  Fun wfun 6348   Fn wfn 6349  wf 6350  cfv 6354  𝑅1cr1 9185  rankcrnk 9186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7574  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-r1 9187  df-rank 9188
This theorem is referenced by:  aomclem5  39542
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