Theorem aomclem5 42719

Description: Lemma for dfac11 42723. 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 479	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 ∈ On)
4		simpr 483	. . . . . 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 479	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
7	1, 3, 4, 6	aomclem4 42718	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → 𝐹 We (𝑅1‘dom 𝑧))
8		iftrue 4539	. . . . . . 7 ⊢ (dom 𝑧 = ∪ dom 𝑧 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐹)
9	8	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐹)
10		eqidd 2727	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
11	9, 10	weeq12d 42701	. . . . 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 256	. . . 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 479	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 ∈ On)
18		eloni 6386	. . . . . . . 8 ⊢ (dom 𝑧 ∈ On → Ord dom 𝑧)
19		orduniorsuc 7839	. . . . . . . 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 1000	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 = suc ∪ dom 𝑧)
22	5	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
23		aomclem5.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
24	23	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → 𝐴 ∈ On)
25		aomclem5.za	. . . . . . 7 ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)
26	25	adantr 479	. . . . . 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 479	. . . . . 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 42717	. . . . 5 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → 𝐸 We (𝑅1‘dom 𝑧))
30		iffalse 4542	. . . . . . 7 ⊢ (¬ dom 𝑧 = ∪ dom 𝑧 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐸)
31	30	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐸)
32		eqidd 2727	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
33	31, 32	weeq12d 42701	. . . . 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 256	. . . 4 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
35	12, 34	pm2.61dan 811	. . 3 ⊢ (𝜑 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
36		weinxp 5766	. . 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 217	. 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 5670	. . 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 233	1 ⊢ (𝜑 → 𝐺 We (𝑅1‘dom 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  Vcvv 3462   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  ∅c0 4325  ifcif 4533  𝒫 cpw 4607  {csn 4633  ∪ cuni 4913  ∩ cint 4954   class class class wbr 5153  {copab 5215   ↦ cmpt 5236   E cep 5585   We wwe 5636   × cxp 5680  ^◡ccnv 5681  dom cdm 5682  ran crn 5683   “ cima 5685  Ord word 6375  Oncon0 6376  suc csuc 6378  ‘cfv 6554  recscrecs 8400  Fincfn 8974  supcsup 9483  𝑅₁cr1 9805  rankcrnk 9806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-map 8857  df-en 8975  df-fin 8978  df-sup 9485  df-r1 9807  df-rank 9808

This theorem is referenced by:  aomclem6  42720