Theorem aomclem3 36644

Description: Lemma for dfac11 36650. 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 5272	. . . . . . 7 ⊢ (𝑎 = 𝑐 → ran 𝑎 = ran 𝑐)
3	2	difeq2d 3690	. . . . . 6 ⊢ (𝑎 = 𝑐 → ((𝑅1‘dom 𝑧) ∖ ran 𝑎) = ((𝑅1‘dom 𝑧) ∖ ran 𝑐))
4	3	fveq2d 6107	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)) = (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
5	4	cbvmptv 4678	. . . 4 ⊢ (𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))) = (𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
6		recseq 7357	. . . 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 2632	. 2 ⊢ 𝐷 = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))))
9		fvex 6113	. . 3 ⊢ (𝑅1‘dom 𝑧) ∈ V
10	9	a1i 11	. 2 ⊢ (𝜑 → (𝑅1‘dom 𝑧) ∈ V)
11		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)))}
12		aomclem3.c	. . . 4 ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))
13		aomclem3.on	. . . 4 ⊢ (𝜑 → dom 𝑧 ∈ On)
14		aomclem3.su	. . . 4 ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)
15		aomclem3.we	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
16		aomclem3.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ On)
17		aomclem3.za	. . . 4 ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)
18		aomclem3.y	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
19	11, 12, 13, 14, 15, 16, 17, 18	aomclem2 36643	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎))
20		neeq1 2844	. . . . 5 ⊢ (𝑎 = 𝑑 → (𝑎 ≠ ∅ ↔ 𝑑 ≠ ∅))
21		fveq2 6103	. . . . . 6 ⊢ (𝑎 = 𝑑 → (𝐶‘𝑎) = (𝐶‘𝑑))
22		id 22	. . . . . 6 ⊢ (𝑎 = 𝑑 → 𝑎 = 𝑑)
23	21, 22	eleq12d 2682	. . . . 5 ⊢ (𝑎 = 𝑑 → ((𝐶‘𝑎) ∈ 𝑎 ↔ (𝐶‘𝑑) ∈ 𝑑))
24	20, 23	imbi12d 333	. . . 4 ⊢ (𝑎 = 𝑑 → ((𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎) ↔ (𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑)))
25	24	cbvralv 3147	. . 3 ⊢ (∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎) ↔ ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑))
26	19, 25	sylib 207	. 2 ⊢ (𝜑 → ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑))
27		aomclem3.e	. 2 ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}
28	8, 10, 26, 27	dnwech 36636	1 ⊢ (𝜑 → 𝐸 We (𝑅1‘dom 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ∩ cint 4410   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   We wwe 4996  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  Oncon0 5640  suc csuc 5642  ‘cfv 5804  recscrecs 7354  Fincfn 7841  supcsup 8229  𝑅₁cr1 8508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-sup 8231  df-r1 8510

This theorem is referenced by:  aomclem5  36646