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Theorem aomclem3 41412
Description: Lemma for dfac11 41418. 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 5896 . . . . . . 7 (𝑎 = 𝑐 → ran 𝑎 = ran 𝑐)
32difeq2d 4087 . . . . . 6 (𝑎 = 𝑐 → ((𝑅1‘dom 𝑧) ∖ ran 𝑎) = ((𝑅1‘dom 𝑧) ∖ ran 𝑐))
43fveq2d 6851 . . . . 5 (𝑎 = 𝑐 → (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)) = (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
54cbvmptv 5223 . . . 4 (𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))) = (𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
6 recseq 8325 . . . 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 2765 . 2 𝐷 = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))))
9 fvexd 6862 . 2 (𝜑 → (𝑅1‘dom 𝑧) ∈ V)
10 aomclem3.b . . . 4 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}
11 aomclem3.c . . . 4 𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (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(𝑧𝑎) We (𝑅1𝑎))
15 aomclem3.a . . . 4 (𝜑𝐴 ∈ On)
16 aomclem3.za . . . 4 (𝜑 → dom 𝑧𝐴)
17 aomclem3.y . . . 4 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
1810, 11, 12, 13, 14, 15, 16, 17aomclem2 41411 . . 3 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶𝑎) ∈ 𝑎))
19 neeq1 3007 . . . . 5 (𝑎 = 𝑑 → (𝑎 ≠ ∅ ↔ 𝑑 ≠ ∅))
20 fveq2 6847 . . . . . 6 (𝑎 = 𝑑 → (𝐶𝑎) = (𝐶𝑑))
21 id 22 . . . . . 6 (𝑎 = 𝑑𝑎 = 𝑑)
2220, 21eleq12d 2832 . . . . 5 (𝑎 = 𝑑 → ((𝐶𝑎) ∈ 𝑎 ↔ (𝐶𝑑) ∈ 𝑑))
2319, 22imbi12d 345 . . . 4 (𝑎 = 𝑑 → ((𝑎 ≠ ∅ → (𝐶𝑎) ∈ 𝑎) ↔ (𝑑 ≠ ∅ → (𝐶𝑑) ∈ 𝑑)))
2423cbvralvw 3228 . . 3 (∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶𝑎) ∈ 𝑎) ↔ ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶𝑑) ∈ 𝑑))
2518, 24sylib 217 . 2 (𝜑 → ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶𝑑) ∈ 𝑑))
26 aomclem3.e . 2 𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}
278, 9, 25, 26dnwech 41404 1 (𝜑𝐸 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  Vcvv 3448  cdif 3912  cin 3914  wss 3915  c0 4287  𝒫 cpw 4565  {csn 4591   cuni 4870   cint 4912   class class class wbr 5110  {copab 5172  cmpt 5193   We wwe 5592  ccnv 5637  dom cdm 5638  ran crn 5639  cima 5641  Oncon0 6322  suc csuc 6324  cfv 6501  recscrecs 8321  Fincfn 8890  supcsup 9383  𝑅1cr1 9705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-map 8774  df-en 8891  df-fin 8894  df-sup 9385  df-r1 9707
This theorem is referenced by:  aomclem5  41414
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