Theorem aomclem5 43075

Description: Lemma for dfac11 43079. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Hypotheses

Ref	Expression
aomclem5.b	⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}
aomclem5.c	⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))
aomclem5.d	⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
aomclem5.e	⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}
aomclem5.f	⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
aomclem5.g	⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
aomclem5.on	⊢ (𝜑 → dom 𝑧 ∈ On)
aomclem5.we	⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
aomclem5.a	⊢ (𝜑 → 𝐴 ∈ On)
aomclem5.za	⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)
aomclem5.y	⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))

Assertion

Ref	Expression
aomclem5	⊢ (𝜑 → 𝐺 We (𝑅1‘dom 𝑧))

Distinct variable groups:   𝑦,𝑧,𝑎,𝑏,𝑐,𝑑   𝜑,𝑎,𝑏   𝐶,𝑎,𝑏,𝑐,𝑑   𝐷,𝑎,𝑏,𝑐,𝑑

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑐,𝑑)   𝐴(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐶(𝑦,𝑧)   𝐷(𝑦,𝑧)   𝐸(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐹(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐺(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem aomclem5

Step	Hyp	Ref	Expression
1		aomclem5.f	. . . . . 6 ⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
2		aomclem5.on	. . . . . . 7 ⊢ (𝜑 → dom 𝑧 ∈ On)
3	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 ∈ On)
4		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 = ∪ dom 𝑧)
5		aomclem5.we	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
7	1, 3, 4, 6	aomclem4 43074	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → 𝐹 We (𝑅1‘dom 𝑧))
8		iftrue 4530	. . . . . . 7 ⊢ (dom 𝑧 = ∪ dom 𝑧 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐹)
9	8	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐹)
10		eqidd 2737	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
11	9, 10	weeq12d 5673	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐹 We (𝑅1‘dom 𝑧)))
12	7, 11	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
13		aomclem5.b	. . . . . 6 ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}
14		aomclem5.c	. . . . . 6 ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))
15		aomclem5.d	. . . . . 6 ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
16		aomclem5.e	. . . . . 6 ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}
17	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 ∈ On)
18		eloni 6393	. . . . . . . 8 ⊢ (dom 𝑧 ∈ On → Ord dom 𝑧)
19		orduniorsuc 7851	. . . . . . . 8 ⊢ (Ord dom 𝑧 → (dom 𝑧 = ∪ dom 𝑧 ∨ dom 𝑧 = suc ∪ dom 𝑧))
20	2, 18, 19	3syl 18	. . . . . . 7 ⊢ (𝜑 → (dom 𝑧 = ∪ dom 𝑧 ∨ dom 𝑧 = suc ∪ dom 𝑧))
21	20	orcanai 1004	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 = suc ∪ dom 𝑧)
22	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
23		aomclem5.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
24	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → 𝐴 ∈ On)
25		aomclem5.za	. . . . . . 7 ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)
26	25	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 ⊆ 𝐴)
27		aomclem5.y	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
28	27	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
29	13, 14, 15, 16, 17, 21, 22, 24, 26, 28	aomclem3 43073	. . . . 5 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → 𝐸 We (𝑅1‘dom 𝑧))
30		iffalse 4533	. . . . . . 7 ⊢ (¬ dom 𝑧 = ∪ dom 𝑧 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐸)
31	30	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐸)
32		eqidd 2737	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
33	31, 32	weeq12d 5673	. . . . 5 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐸 We (𝑅1‘dom 𝑧)))
34	29, 33	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
35	12, 34	pm2.61dan 812	. . 3 ⊢ (𝜑 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
36		weinxp 5769	. . 3 ⊢ (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧))
37	35, 36	sylib 218	. 2 ⊢ (𝜑 → (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧))
38		aomclem5.g	. . 3 ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
39		weeq1 5671	. . 3 ⊢ (𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) → (𝐺 We (𝑅1‘dom 𝑧) ↔ (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧)))
40	38, 39	ax-mp 5	. 2 ⊢ (𝐺 We (𝑅1‘dom 𝑧) ↔ (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧))
41	37, 40	sylibr 234	1 ⊢ (𝜑 → 𝐺 We (𝑅1‘dom 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ∖ cdif 3947   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  ifcif 4524  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906  ∩ cint 4945   class class class wbr 5142  {copab 5204   ↦ cmpt 5224   E cep 5582   We wwe 5635   × cxp 5682  ^◡ccnv 5683  dom cdm 5684  ran crn 5685   “ cima 5687  Ord word 6382  Oncon0 6383  suc csuc 6385  ‘cfv 6560  recscrecs 8411  Fincfn 8986  supcsup 9481  𝑅₁cr1 9803  rankcrnk 9804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-map 8869  df-en 8987  df-fin 8990  df-sup 9483  df-r1 9805  df-rank 9806

This theorem is referenced by:  aomclem6  43076