Theorem aomclem3 44167

Description: Lemma for dfac11 44173. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Hypotheses

Ref	Expression
aomclem3.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)))}
aomclem3.c	⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))
aomclem3.d	⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
aomclem3.e	⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}
aomclem3.on	⊢ (𝜑 → dom 𝑧 ∈ On)
aomclem3.su	⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)
aomclem3.we	⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
aomclem3.a	⊢ (𝜑 → 𝐴 ∈ On)
aomclem3.za	⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)
aomclem3.y	⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))

Assertion

Ref	Expression
aomclem3	⊢ (𝜑 → 𝐸 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)

Proof of Theorem aomclem3

Step	Hyp	Ref	Expression
1		aomclem3.d	. . 3 ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
2		rneq 5919	. . . . . . 7 ⊢ (𝑎 = 𝑐 → ran 𝑎 = ran 𝑐)
3	2	difeq2d 4062	. . . . . 6 ⊢ (𝑎 = 𝑐 → ((𝑅1‘dom 𝑧) ∖ ran 𝑎) = ((𝑅1‘dom 𝑧) ∖ ran 𝑐))
4	3	fveq2d 6894	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)) = (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
5	4	cbvmptv 5208	. . . 4 ⊢ (𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))) = (𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
6		recseq 8388	. . . 4 ⊢ ((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))) = (𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))) → recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))))
7	5, 6	ax-mp 5	. . 3 ⊢ recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))))
8	1, 7	eqtri 2783	. 2 ⊢ 𝐷 = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))))
9		fvexd 6905	. 2 ⊢ (𝜑 → (𝑅1‘dom 𝑧) ∈ V)
10		aomclem3.b	. . . 4 ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}
11		aomclem3.c	. . . 4 ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))
12		aomclem3.on	. . . 4 ⊢ (𝜑 → dom 𝑧 ∈ On)
13		aomclem3.su	. . . 4 ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)
14		aomclem3.we	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
15		aomclem3.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ On)
16		aomclem3.za	. . . 4 ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)
17		aomclem3.y	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
18	10, 11, 12, 13, 14, 15, 16, 17	aomclem2 44166	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎))
19		neeq1 3017	. . . . 5 ⊢ (𝑎 = 𝑑 → (𝑎 ≠ ∅ ↔ 𝑑 ≠ ∅))
20		fveq2 6890	. . . . . 6 ⊢ (𝑎 = 𝑑 → (𝐶‘𝑎) = (𝐶‘𝑑))
21		id 23	. . . . . 6 ⊢ (𝑎 = 𝑑 → 𝑎 = 𝑑)
22	20, 21	eleq12d 2854	. . . . 5 ⊢ (𝑎 = 𝑑 → ((𝐶‘𝑎) ∈ 𝑎 ↔ (𝐶‘𝑑) ∈ 𝑑))
23	19, 22	imbi12d 347	. . . 4 ⊢ (𝑎 = 𝑑 → ((𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎) ↔ (𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑)))
24	23	cbvralvw 3240	. . 3 ⊢ (∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎) ↔ ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑))
25	18, 24	sylib 221	. 2 ⊢ (𝜑 → ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑))
26		aomclem3.e	. 2 ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}
27	8, 9, 25, 26	dnwech 44164	1 ⊢ (𝜑 → 𝐸 We (𝑅1‘dom 𝑧))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044   ∩ cin 4088  ∅c0 4277  𝒫 cpw 4556  {csn 4584  ∪ cuni 4866  ∩ cint 4906   class class class wbr 5102  {copab 5166   ↦ cmpt 5185   We wwe 5603  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   “ cima 5654  Oncon0 6365  suc csuc 6367  ‘cfv 6543  recscrecs 8385  Fincfn 8980  supcsup 9439  𝑅₁cr1 9773

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-map 8856  df-en 8981  df-fin 8984  df-sup 9441  df-r1 9775

This theorem is used by:  aomclem5  44169