Theorem aomclem5 40799

Description: Lemma for dfac11 40803. 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 40798	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → 𝐹 We (𝑅1‘dom 𝑧))
8		iftrue 4462	. . . . . . 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 2739	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
11	9, 10	weeq12d 40781	. . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 ∈ On)
18		eloni 6261	. . . . . . . 8 ⊢ (dom 𝑧 ∈ On → Ord dom 𝑧)
19		orduniorsuc 7652	. . . . . . . 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 999	. . . . . 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 40797	. . . . 5 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → 𝐸 We (𝑅1‘dom 𝑧))
30		iffalse 4465	. . . . . . 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 2739	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
33	31, 32	weeq12d 40781	. . . . 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 809	. . 3 ⊢ (𝜑 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
36		weinxp 5662	. . 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 5568	. . 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 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ifcif 4456  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836  ∩ cint 4876   class class class wbr 5070  {copab 5132   ↦ cmpt 5153   E cep 5485   We wwe 5534   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583  Ord word 6250  Oncon0 6251  suc csuc 6253  ‘cfv 6418  recscrecs 8172  Fincfn 8691  supcsup 9129  𝑅₁cr1 9451  rankcrnk 9452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-map 8575  df-en 8692  df-fin 8695  df-sup 9131  df-r1 9453  df-rank 9454

This theorem is referenced by:  aomclem6  40800