Theorem aomclem4 39655

Description: Lemma for dfac11 39660. Limit case. Patch together well-orderings constructed so far using fnwe2 39651 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(𝑧‘𝑎) We (𝑅1‘𝑎))

Assertion

Ref	Expression
aomclem4	⊢ (𝜑 → 𝐹 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.

Step	Hyp	Ref	Expression
1		suceq 6255	. . 3 ⊢ (𝑐 = (rank‘𝑎) → suc 𝑐 = suc (rank‘𝑎))
2	1	fveq2d 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 9195	. . . . . . . . . . . . . 14 ⊢ 𝑅1 Fn On
5		fnfun 6452	. . . . . . . . . . . . . 14 ⊢ (𝑅1 Fn On → Fun 𝑅1)
6	4, 5	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 𝑅1
7		fndm 6454	. . . . . . . . . . . . . . 15 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
8	4, 7	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom 𝑅1 = On
9	8	eqimss2i 4025	. . . . . . . . . . . . 13 ⊢ On ⊆ dom 𝑅1
10	6, 9	pm3.2i 473	. . . . . . . . . . . 12 ⊢ (Fun 𝑅1 ∧ On ⊆ dom 𝑅1)
11		aomclem4.on	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑧 ∈ On)
12		funfvima2 6992	. . . . . . . . . . . 12 ⊢ ((Fun 𝑅1 ∧ On ⊆ dom 𝑅1) → (dom 𝑧 ∈ On → (𝑅1‘dom 𝑧) ∈ (𝑅1 “ On)))
13	10, 11, 12	mpsyl 68	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅1‘dom 𝑧) ∈ (𝑅1 “ On))
14		elssuni 4867	. . . . . . . . . . 11 ⊢ ((𝑅1‘dom 𝑧) ∈ (𝑅1 “ On) → (𝑅1‘dom 𝑧) ⊆ ∪ (𝑅1 “ On))
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅1‘dom 𝑧) ⊆ ∪ (𝑅1 “ On))
16	15	sselda 3966	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 ∈ ∪ (𝑅1 “ On))
17		rankidb 9228	. . . . . . . . 9 ⊢ (𝑏 ∈ ∪ (𝑅1 “ On) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
18	16, 17	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
19		suceq 6255	. . . . . . . . . 10 ⊢ ((rank‘𝑏) = (rank‘𝑎) → suc (rank‘𝑏) = suc (rank‘𝑎))
20	19	fveq2d 6673	. . . . . . . . 9 ⊢ ((rank‘𝑏) = (rank‘𝑎) → (𝑅1‘suc (rank‘𝑏)) = (𝑅1‘suc (rank‘𝑎)))
21	20	eleq2d 2898	. . . . . . . 8 ⊢ ((rank‘𝑏) = (rank‘𝑎) → (𝑏 ∈ (𝑅1‘suc (rank‘𝑏)) ↔ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
22	18, 21	syl5ibcom 247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑏) = (rank‘𝑎) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
23	22	expimpd 456	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
24	23	ss2abdv 4043	. . . . 5 ⊢ (𝜑 → {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))} ⊆ {𝑏 ∣ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))})
25		df-rab 3147	. . . . 5 ⊢ {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} = {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))}
26		abid1 2956	. . . . 5 ⊢ (𝑅1‘suc (rank‘𝑎)) = {𝑏 ∣ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))}
27	24, 25, 26	3sstr4g 4011	. . . 4 ⊢ (𝜑 → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
28	27	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
29		fveq2 6669	. . . . 5 ⊢ (𝑏 = suc (rank‘𝑎) → (𝑧‘𝑏) = (𝑧‘suc (rank‘𝑎)))
30		fveq2 6669	. . . . 5 ⊢ (𝑏 = suc (rank‘𝑎) → (𝑅1‘𝑏) = (𝑅1‘suc (rank‘𝑎)))
31	29, 30	weeq12d 39638	. . . 4 ⊢ (𝑏 = suc (rank‘𝑎) → ((𝑧‘𝑏) We (𝑅1‘𝑏) ↔ (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎))))
32		aomclem4.we	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
33		fveq2 6669	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑧‘𝑎) = (𝑧‘𝑏))
34		fveq2 6669	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑅1‘𝑎) = (𝑅1‘𝑏))
35	33, 34	weeq12d 39638	. . . . . . 7 ⊢ (𝑎 = 𝑏 → ((𝑧‘𝑎) We (𝑅1‘𝑎) ↔ (𝑧‘𝑏) We (𝑅1‘𝑏)))
36	35	cbvralvw 3449	. . . . . 6 ⊢ (∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎) ↔ ∀𝑏 ∈ dom 𝑧(𝑧‘𝑏) We (𝑅1‘𝑏))
37	32, 36	sylib 220	. . . . 5 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑧(𝑧‘𝑏) We (𝑅1‘𝑏))
38	37	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → ∀𝑏 ∈ dom 𝑧(𝑧‘𝑏) We (𝑅1‘𝑏))
39		rankr1ai 9226	. . . . . 6 ⊢ (𝑎 ∈ (𝑅1‘dom 𝑧) → (rank‘𝑎) ∈ dom 𝑧)
40	39	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → (rank‘𝑎) ∈ dom 𝑧)
41		eloni 6200	. . . . . . . 8 ⊢ (dom 𝑧 ∈ On → Ord dom 𝑧)
42	11, 41	syl 17	. . . . . . 7 ⊢ (𝜑 → Ord dom 𝑧)
43		aomclem4.su	. . . . . . 7 ⊢ (𝜑 → dom 𝑧 = ∪ dom 𝑧)
44		limsuc2 39639	. . . . . . 7 ⊢ ((Ord dom 𝑧 ∧ dom 𝑧 = ∪ dom 𝑧) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
45	42, 43, 44	syl2anc 586	. . . . . 6 ⊢ (𝜑 → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
46	45	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
47	40, 46	mpbid 234	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → suc (rank‘𝑎) ∈ dom 𝑧)
48	31, 38, 47	rspcdva 3624	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (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)}))
50	28, 48, 49	sylc 65	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)})
51		rankf 9222	. . . 4 ⊢ rank:∪ (𝑅1 “ On)⟶On
52	51	a1i 11	. . 3 ⊢ (𝜑 → rank:∪ (𝑅1 “ On)⟶On)
53	52, 15	fssresd 6544	. 2 ⊢ (𝜑 → (rank ↾ (𝑅1‘dom 𝑧)):(𝑅1‘dom 𝑧)⟶On)
54		epweon 7496	. . 3 ⊢ E We On
55	54	a1i 11	. 2 ⊢ (𝜑 → E We On)
56	2, 3, 50, 53, 55	fnwe2 39651	1 ⊢ (𝜑 → 𝐹 We (𝑅1‘dom 𝑧))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  {crab 3142   ⊆ wss 3935  ∪ cuni 4837   class class class wbr 5065  {copab 5127   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 9190  rankcrnk 9191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  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 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-r1 9192  df-rank 9193

This theorem is referenced by:  aomclem5  39656