Theorem aomclem3 37626

Description: Lemma for dfac11 37632. 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 5351	. . . . . . 7 ⊢ (𝑎 = 𝑐 → ran 𝑎 = ran 𝑐)
3	2	difeq2d 3728	. . . . . 6 ⊢ (𝑎 = 𝑐 → ((𝑅1‘dom 𝑧) ∖ ran 𝑎) = ((𝑅1‘dom 𝑧) ∖ ran 𝑐))
4	3	fveq2d 6195	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)) = (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
5	4	cbvmptv 4750	. . . 4 ⊢ (𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))) = (𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
6		recseq 7470	. . . 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 2644	. 2 ⊢ 𝐷 = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))))
9		fvexd 6203	. 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 37625	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎))
19		neeq1 2856	. . . . 5 ⊢ (𝑎 = 𝑑 → (𝑎 ≠ ∅ ↔ 𝑑 ≠ ∅))
20		fveq2 6191	. . . . . 6 ⊢ (𝑎 = 𝑑 → (𝐶‘𝑎) = (𝐶‘𝑑))
21		id 22	. . . . . 6 ⊢ (𝑎 = 𝑑 → 𝑎 = 𝑑)
22	20, 21	eleq12d 2695	. . . . 5 ⊢ (𝑎 = 𝑑 → ((𝐶‘𝑎) ∈ 𝑎 ↔ (𝐶‘𝑑) ∈ 𝑑))
23	19, 22	imbi12d 334	. . . 4 ⊢ (𝑎 = 𝑑 → ((𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎) ↔ (𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑)))
24	23	cbvralv 3171	. . 3 ⊢ (∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎) ↔ ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑))
25	18, 24	sylib 208	. 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 37618	1 ⊢ (𝜑 → 𝐸 We (𝑅1‘dom 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  ∩ cint 4475   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   We wwe 5072  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117  Oncon0 5723  suc csuc 5725  ‘cfv 5888  recscrecs 7467  Fincfn 7955  supcsup 8346  𝑅₁cr1 8625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-sup 8348  df-r1 8627

This theorem is referenced by:  aomclem5  37628