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Theorem aomclem3 37103
Description: Lemma for dfac11 37109. 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)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}
aomclem3.c 𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (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(𝑧𝑎) We (𝑅1𝑎))
aomclem3.a (𝜑𝐴 ∈ On)
aomclem3.za (𝜑 → dom 𝑧𝐴)
aomclem3.y (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
Assertion
Ref Expression
aomclem3 (𝜑𝐸 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
StepHypRef Expression
1 aomclem3.d . . 3 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
2 rneq 5311 . . . . . . 7 (𝑎 = 𝑐 → ran 𝑎 = ran 𝑐)
32difeq2d 3706 . . . . . 6 (𝑎 = 𝑐 → ((𝑅1‘dom 𝑧) ∖ ran 𝑎) = ((𝑅1‘dom 𝑧) ∖ ran 𝑐))
43fveq2d 6152 . . . . 5 (𝑎 = 𝑐 → (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)) = (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
54cbvmptv 4710 . . . 4 (𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))) = (𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
6 recseq 7415 . . . 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)))))
75, 6ax-mp 5 . . 3 recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))))
81, 7eqtri 2643 . 2 𝐷 = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))))
9 fvex 6158 . . 3 (𝑅1‘dom 𝑧) ∈ V
109a1i 11 . 2 (𝜑 → (𝑅1‘dom 𝑧) ∈ V)
11 aomclem3.b . . . 4 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}
12 aomclem3.c . . . 4 𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (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(𝑧𝑎) We (𝑅1𝑎))
16 aomclem3.a . . . 4 (𝜑𝐴 ∈ On)
17 aomclem3.za . . . 4 (𝜑 → dom 𝑧𝐴)
18 aomclem3.y . . . 4 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
1911, 12, 13, 14, 15, 16, 17, 18aomclem2 37102 . . 3 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶𝑎) ∈ 𝑎))
20 neeq1 2852 . . . . 5 (𝑎 = 𝑑 → (𝑎 ≠ ∅ ↔ 𝑑 ≠ ∅))
21 fveq2 6148 . . . . . 6 (𝑎 = 𝑑 → (𝐶𝑎) = (𝐶𝑑))
22 id 22 . . . . . 6 (𝑎 = 𝑑𝑎 = 𝑑)
2321, 22eleq12d 2692 . . . . 5 (𝑎 = 𝑑 → ((𝐶𝑎) ∈ 𝑎 ↔ (𝐶𝑑) ∈ 𝑑))
2420, 23imbi12d 334 . . . 4 (𝑎 = 𝑑 → ((𝑎 ≠ ∅ → (𝐶𝑎) ∈ 𝑎) ↔ (𝑑 ≠ ∅ → (𝐶𝑑) ∈ 𝑑)))
2524cbvralv 3159 . . 3 (∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶𝑎) ∈ 𝑎) ↔ ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶𝑑) ∈ 𝑑))
2619, 25sylib 208 . 2 (𝜑 → ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶𝑑) ∈ 𝑑))
27 aomclem3.e . 2 𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}
288, 10, 26, 27dnwech 37095 1 (𝜑𝐸 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   cuni 4402   cint 4440   class class class wbr 4613  {copab 4672  cmpt 4673   We wwe 5032  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  Oncon0 5682  suc csuc 5684  cfv 5847  recscrecs 7412  Fincfn 7899  supcsup 8290  𝑅1cr1 8569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-er 7687  df-map 7804  df-en 7900  df-fin 7903  df-sup 8292  df-r1 8571
This theorem is referenced by:  aomclem5  37105
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