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Theorem aomclem3 38128
Description: Lemma for dfac11 38134. 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 5506 . . . . . . 7 (𝑎 = 𝑐 → ran 𝑎 = ran 𝑐)
32difeq2d 3871 . . . . . 6 (𝑎 = 𝑐 → ((𝑅1‘dom 𝑧) ∖ ran 𝑎) = ((𝑅1‘dom 𝑧) ∖ ran 𝑐))
43fveq2d 6356 . . . . 5 (𝑎 = 𝑐 → (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)) = (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
54cbvmptv 4902 . . . 4 (𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))) = (𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))
6 recseq 7639 . . . 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 2782 . 2 𝐷 = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))))
9 fvexd 6364 . 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 38127 . . 3 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶𝑎) ∈ 𝑎))
19 neeq1 2994 . . . . 5 (𝑎 = 𝑑 → (𝑎 ≠ ∅ ↔ 𝑑 ≠ ∅))
20 fveq2 6352 . . . . . 6 (𝑎 = 𝑑 → (𝐶𝑎) = (𝐶𝑑))
21 id 22 . . . . . 6 (𝑎 = 𝑑𝑎 = 𝑑)
2220, 21eleq12d 2833 . . . . 5 (𝑎 = 𝑑 → ((𝐶𝑎) ∈ 𝑎 ↔ (𝐶𝑑) ∈ 𝑑))
2319, 22imbi12d 333 . . . 4 (𝑎 = 𝑑 → ((𝑎 ≠ ∅ → (𝐶𝑎) ∈ 𝑎) ↔ (𝑑 ≠ ∅ → (𝐶𝑑) ∈ 𝑑)))
2423cbvralv 3310 . . 3 (∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶𝑎) ∈ 𝑎) ↔ ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶𝑑) ∈ 𝑑))
2518, 24sylib 208 . 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 38120 1 (𝜑𝐸 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321   cuni 4588   cint 4627   class class class wbr 4804  {copab 4864  cmpt 4881   We wwe 5224  ccnv 5265  dom cdm 5266  ran crn 5267  cima 5269  Oncon0 5884  suc csuc 5886  cfv 6049  recscrecs 7636  Fincfn 8121  supcsup 8511  𝑅1cr1 8798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-er 7911  df-map 8025  df-en 8122  df-fin 8125  df-sup 8513  df-r1 8800
This theorem is referenced by:  aomclem5  38130
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