Theorem aomclem4 41785

Description: Lemma for dfac11 41790. Limit case. Patch together well-orderings constructed so far using fnwe2 41781 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 6428	. . 3 ⊢ (𝑐 = (rank‘𝑎) → suc 𝑐 = suc (rank‘𝑎))
2	1	fveq2d 6893	. 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 9759	. . . . . . . . . . . . . 14 ⊢ 𝑅1 Fn On
5		fnfun 6647	. . . . . . . . . . . . . 14 ⊢ (𝑅1 Fn On → Fun 𝑅1)
6	4, 5	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 𝑅1
7	4	fndmi 6651	. . . . . . . . . . . . . 14 ⊢ dom 𝑅1 = On
8	7	eqimss2i 4043	. . . . . . . . . . . . 13 ⊢ On ⊆ dom 𝑅1
9	6, 8	pm3.2i 472	. . . . . . . . . . . 12 ⊢ (Fun 𝑅1 ∧ On ⊆ dom 𝑅1)
10		aomclem4.on	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑧 ∈ On)
11		funfvima2 7230	. . . . . . . . . . . 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 4941	. . . . . . . . . . 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 3982	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝑅1‘dom 𝑧)) → 𝑏 ∈ ∪ (𝑅1 “ On))
16		rankidb 9792	. . . . . . . . 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 6428	. . . . . . . . . 10 ⊢ ((rank‘𝑏) = (rank‘𝑎) → suc (rank‘𝑏) = suc (rank‘𝑎))
19	18	fveq2d 6893	. . . . . . . . 9 ⊢ ((rank‘𝑏) = (rank‘𝑎) → (𝑅1‘suc (rank‘𝑏)) = (𝑅1‘suc (rank‘𝑎)))
20	19	eleq2d 2820	. . . . . . . 8 ⊢ ((rank‘𝑏) = (rank‘𝑎) → (𝑏 ∈ (𝑅1‘suc (rank‘𝑏)) ↔ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
21	17, 20	syl5ibcom 244	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑏) = (rank‘𝑎) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
22	21	expimpd 455	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎)) → 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))))
23	22	ss2abdv 4060	. . . . 5 ⊢ (𝜑 → {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))} ⊆ {𝑏 ∣ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))})
24		df-rab 3434	. . . . 5 ⊢ {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} = {𝑏 ∣ (𝑏 ∈ (𝑅1‘dom 𝑧) ∧ (rank‘𝑏) = (rank‘𝑎))}
25		abid1 2871	. . . . 5 ⊢ (𝑅1‘suc (rank‘𝑎)) = {𝑏 ∣ 𝑏 ∈ (𝑅1‘suc (rank‘𝑎))}
26	23, 24, 25	3sstr4g 4027	. . . 4 ⊢ (𝜑 → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
27	26	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → {𝑏 ∈ (𝑅1‘dom 𝑧) ∣ (rank‘𝑏) = (rank‘𝑎)} ⊆ (𝑅1‘suc (rank‘𝑎)))
28		fveq2 6889	. . . . 5 ⊢ (𝑏 = suc (rank‘𝑎) → (𝑧‘𝑏) = (𝑧‘suc (rank‘𝑎)))
29		fveq2 6889	. . . . 5 ⊢ (𝑏 = suc (rank‘𝑎) → (𝑅1‘𝑏) = (𝑅1‘suc (rank‘𝑎)))
30	28, 29	weeq12d 41768	. . . 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 6889	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑧‘𝑎) = (𝑧‘𝑏))
33		fveq2 6889	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑅1‘𝑎) = (𝑅1‘𝑏))
34	32, 33	weeq12d 41768	. . . . . . 7 ⊢ (𝑎 = 𝑏 → ((𝑧‘𝑎) We (𝑅1‘𝑎) ↔ (𝑧‘𝑏) We (𝑅1‘𝑏)))
35	34	cbvralvw 3235	. . . . . 6 ⊢ (∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎) ↔ ∀𝑏 ∈ dom 𝑧(𝑧‘𝑏) We (𝑅1‘𝑏))
36	31, 35	sylib 217	. . . . 5 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑧(𝑧‘𝑏) We (𝑅1‘𝑏))
37	36	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → ∀𝑏 ∈ dom 𝑧(𝑧‘𝑏) We (𝑅1‘𝑏))
38		rankr1ai 9790	. . . . . 6 ⊢ (𝑎 ∈ (𝑅1‘dom 𝑧) → (rank‘𝑎) ∈ dom 𝑧)
39	38	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → (rank‘𝑎) ∈ dom 𝑧)
40		eloni 6372	. . . . . . . 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 41769	. . . . . . 7 ⊢ ((Ord dom 𝑧 ∧ dom 𝑧 = ∪ dom 𝑧) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
44	41, 42, 43	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
45	44	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → ((rank‘𝑎) ∈ dom 𝑧 ↔ suc (rank‘𝑎) ∈ dom 𝑧))
46	39, 45	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → suc (rank‘𝑎) ∈ dom 𝑧)
47	30, 37, 46	rspcdva 3614	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑅1‘dom 𝑧)) → (𝑧‘suc (rank‘𝑎)) We (𝑅1‘suc (rank‘𝑎)))
48		wess 5663	. . 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 9786	. . . 4 ⊢ rank:∪ (𝑅1 “ On)⟶On
51	50	a1i 11	. . 3 ⊢ (𝜑 → rank:∪ (𝑅1 “ On)⟶On)
52	51, 14	fssresd 6756	. 2 ⊢ (𝜑 → (rank ↾ (𝑅1‘dom 𝑧)):(𝑅1‘dom 𝑧)⟶On)
53		epweon 7759	. . 3 ⊢ E We On
54	53	a1i 11	. 2 ⊢ (𝜑 → E We On)
55	2, 3, 49, 52, 54	fnwe2 41781	1 ⊢ (𝜑 → 𝐹 We (𝑅1‘dom 𝑧))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  {crab 3433   ⊆ wss 3948  ∪ cuni 4908   class class class wbr 5148  {copab 5210   E cep 5579   We wwe 5630  dom cdm 5676   “ cima 5679  Ord word 6361  Oncon0 6362  suc csuc 6364  Fun wfun 6535   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  𝑅₁cr1 9754  rankcrnk 9755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-om 7853  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-r1 9756  df-rank 9757

This theorem is referenced by:  aomclem5  41786