Theorem aomclem5 43805

Description: Lemma for dfac11 43809. 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 485	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 ∈ On)
4		simpr 489	. . . . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
7	1, 3, 4, 6	aomclem4 43804	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → 𝐹 We (𝑅1‘dom 𝑧))
8		iftrue 4493	. . . . . . 7 ⊢ (dom 𝑧 = ∪ dom 𝑧 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐹)
9	8	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐹)
10		eqidd 2764	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
11	9, 10	weeq12d 5650	. . . . 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 260	. . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 ∈ On)
18		eloni 6370	. . . . . . . 8 ⊢ (dom 𝑧 ∈ On → Ord dom 𝑧)
19		orduniorsuc 7822	. . . . . . . 8 ⊢ (Ord dom 𝑧 → (dom 𝑧 = ∪ dom 𝑧 ∨ dom 𝑧 = suc ∪ dom 𝑧))
20	2, 18, 19	3syl 19	. . . . . . 7 ⊢ (𝜑 → (dom 𝑧 = ∪ dom 𝑧 ∨ dom 𝑧 = suc ∪ dom 𝑧))
21	20	orcanai 1018	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 = suc ∪ dom 𝑧)
22	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
23		aomclem5.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
24	23	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → 𝐴 ∈ On)
25		aomclem5.za	. . . . . . 7 ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)
26	25	adantr 485	. . . . . 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 485	. . . . . 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 43803	. . . . 5 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → 𝐸 We (𝑅1‘dom 𝑧))
30		iffalse 4496	. . . . . . 7 ⊢ (¬ dom 𝑧 = ∪ dom 𝑧 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐸)
31	30	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐸)
32		eqidd 2764	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
33	31, 32	weeq12d 5650	. . . . 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 260	. . . 4 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
35	12, 34	pm2.61dan 824	. . 3 ⊢ (𝜑 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
36		weinxp 5746	. . 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 221	. 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 5648	. . 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 237	1 ⊢ (𝜑 → 𝐺 We (𝑅1‘dom 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  ifcif 4487  𝒫 cpw 4562  {csn 4589  ∪ cuni 4872  ∩ cint 4912   class class class wbr 5109  {copab 5173   ↦ cmpt 5192   E cep 5560   We wwe 5613   × cxp 5659  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   “ cima 5664  Ord word 6359  Oncon0 6360  suc csuc 6362  ‘cfv 6536  recscrecs 8353  Fincfn 8939  supcsup 9396  𝑅₁cr1 9730  rankcrnk 9731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-map 8822  df-en 8940  df-fin 8943  df-sup 9398  df-r1 9732  df-rank 9733

This theorem is referenced by:  aomclem6  43806