Theorem aomclem4 43014

Description: Lemma for dfac11 43019. Limit case. Patch together well-orderings constructed so far using fnwe2 43010 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 6461	. . 3 ⊢ (𝑐 = (rank‘𝑎) → suc 𝑐 = suc (rank‘𝑎))
2	1	fveq2d 6924	. 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 9836	. . . . . . . . . . . . . 14 ⊢ 𝑅1 Fn On
5		fnfun 6679	. . . . . . . . . . . . . 14 ⊢ (𝑅1 Fn On → Fun 𝑅1)
6	4, 5	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 𝑅1
7	4	fndmi 6683	. . . . . . . . . . . . . 14 ⊢ dom 𝑅1 = On
8	7	eqimss2i 4070	. . . . . . . . . . . . 13 ⊢ On ⊆ dom 𝑅1
9	6, 8	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (Fun 𝑅1 ∧ On ⊆ dom 𝑅1)
10		aomclem4.on	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑧 ∈ On)
11		funfvima2 7268	. . . . . . . . . . . 12 ⊢ ((Fun 𝑅1 ∧ On ⊆ dom 𝑅1) → (dom 𝑧 ∈ On → (𝑅1‘dom 𝑧) ∈ (𝑅1 “ On)))
12	9, 10, 11	mpsyl 68	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅1‘dom 𝑧) ∈ (𝑅1 “ On))
13		elssuni 4961	. . . . . . . . . . 11 ⊢ ((𝑅1‘dom 𝑧) ∈ (𝑅1 “ On) → (𝑅1‘dom 𝑧) ⊆ ∪ (𝑅1 “ On))
14	12, 13	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅1‘dom 𝑧) ⊆ ∪ (𝑅1 “ On))
15	14	sselda 4008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 ∈ ∪ (𝑅1 “ On))
16		rankidb 9869	. . . . . . . . 9 ⊢ (𝑏 ∈ ∪ (𝑅1 “ On) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
17	15, 16	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑏)))
18		suceq 6461	. . . . . . . . . 10 ⊢ ((rank‘𝑏) = (rank‘𝑎) → suc (rank‘𝑏) = suc (rank‘𝑎))
19	18	fveq2d 6924	. . . . . . . . 9 ⊢ ((rank‘𝑏) = (rank‘𝑎) → (𝑅1‘suc (rank‘𝑏)) = (𝑅1‘suc (rank‘𝑎)))
20	19	eleq2d 2830	. . . . . . . 8 ⊢ ((rank‘𝑏) = (rank‘𝑎) → (𝑏 ∈ (𝑅1‘suc (rank‘𝑏)) ↔ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
21	17, 20	syl5ibcom 245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑏) = (rank‘𝑎) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
22	21	expimpd 453	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
23	22	ss2abdv 4089	. . . . 5 ⊢ (𝜑 → {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))} ⊆ {𝑏 ∣ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))})
24		df-rab 3444	. . . . 5 ⊢ {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} = {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))}
25		abid1 2881	. . . . 5 ⊢ (𝑅1‘suc (rank‘𝑎)) = {𝑏 ∣ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))}
26	23, 24, 25	3sstr4g 4054	. . . 4 ⊢ (𝜑 → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
27	26	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
28		fveq2 6920	. . . . 5 ⊢ (𝑏 = suc (rank‘𝑎) → (𝑧‘𝑏) = (𝑧‘suc (rank‘𝑎)))
29		fveq2 6920	. . . . 5 ⊢ (𝑏 = suc (rank‘𝑎) → (𝑅1‘𝑏) = (𝑅1‘suc (rank‘𝑎)))
30	28, 29	weeq12d 5689	. . . 4 ⊢ (𝑏 = suc (rank‘𝑎) → ((𝑧‘𝑏) We (𝑅1‘𝑏) ↔ (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎))))
31		aomclem4.we	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
32		fveq2 6920	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑧‘𝑎) = (𝑧‘𝑏))
33		fveq2 6920	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑅1‘𝑎) = (𝑅1‘𝑏))
34	32, 33	weeq12d 5689	. . . . . . 7 ⊢ (𝑎 = 𝑏 → ((𝑧‘𝑎) We (𝑅1‘𝑎) ↔ (𝑧‘𝑏) We (𝑅1‘𝑏)))
35	34	cbvralvw 3243	. . . . . 6 ⊢ (∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎) ↔ ∀𝑏 ∈ dom 𝑧(𝑧‘𝑏) We (𝑅1‘𝑏))
36	31, 35	sylib 218	. . . . 5 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑧(𝑧‘𝑏) We (𝑅1‘𝑏))
37	36	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → ∀𝑏 ∈ dom 𝑧(𝑧‘𝑏) We (𝑅1‘𝑏))
38		rankr1ai 9867	. . . . . 6 ⊢ (𝑎 ∈ (𝑅1‘dom 𝑧) → (rank‘𝑎) ∈ dom 𝑧)
39	38	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → (rank‘𝑎) ∈ dom 𝑧)
40		eloni 6405	. . . . . . . 8 ⊢ (dom 𝑧 ∈ On → Ord dom 𝑧)
41	10, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → Ord dom 𝑧)
42		aomclem4.su	. . . . . . 7 ⊢ (𝜑 → dom 𝑧 = ∪ dom 𝑧)
43		limsuc2 42998	. . . . . . 7 ⊢ ((Ord dom 𝑧 ∧ dom 𝑧 = ∪ dom 𝑧) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
44	41, 42, 43	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
45	44	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
46	39, 45	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → suc (rank‘𝑎) ∈ dom 𝑧)
47	30, 37, 46	rspcdva 3636	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎)))
48		wess 5686	. . 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)}))
49	27, 47, 48	sylc 65	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)})
50		rankf 9863	. . . 4 ⊢ rank:∪ (𝑅1 “ On)⟶On
51	50	a1i 11	. . 3 ⊢ (𝜑 → rank:∪ (𝑅1 “ On)⟶On)
52	51, 14	fssresd 6788	. 2 ⊢ (𝜑 → (rank ↾ (𝑅1‘dom 𝑧)):(𝑅1‘dom 𝑧)⟶On)
53		epweon 7810	. . 3 ⊢ E We On
54	53	a1i 11	. 2 ⊢ (𝜑 → E We On)
55	2, 3, 49, 52, 54	fnwe2 43010	1 ⊢ (𝜑 → 𝐹 We (𝑅1‘dom 𝑧))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  {crab 3443   ⊆ wss 3976  ∪ cuni 4931   class class class wbr 5166  {copab 5228   E cep 5598   We wwe 5651  dom cdm 5700   “ cima 5703  Ord word 6394  Oncon0 6395  suc csuc 6397  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  𝑅₁cr1 9831  rankcrnk 9832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-r1 9833  df-rank 9834

This theorem is referenced by:  aomclem5  43015