Theorem aomclem7 43217

Description: Lemma for dfac11 43219. (𝑅₁‘𝐴) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Hypotheses

Ref	Expression
aomclem6.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)))}
aomclem6.c	⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))
aomclem6.d	⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
aomclem6.e	⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}
aomclem6.f	⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
aomclem6.g	⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
aomclem6.h	⊢ 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))
aomclem6.a	⊢ (𝜑 → 𝐴 ∈ On)
aomclem6.y	⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))

Assertion

Ref	Expression
aomclem7	⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴))

Distinct variable groups:   𝑦,𝑧,𝑎,𝑏,𝑐,𝑑   𝜑,𝑎,𝑏,𝑐,𝑑,𝑧   𝐶,𝑎,𝑏,𝑐,𝑑   𝐷,𝑎,𝑏,𝑐,𝑑   𝐴,𝑎,𝑏,𝑐,𝑑,𝑧   𝐻,𝑎,𝑏,𝑐,𝑑,𝑧   𝐺,𝑑

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐶(𝑦,𝑧)   𝐷(𝑦,𝑧)   𝐸(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐹(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐺(𝑦,𝑧,𝑎,𝑏,𝑐)   𝐻(𝑦)

Proof of Theorem aomclem7

Step	Hyp	Ref	Expression
1		aomclem6.b	. . 3 ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}
2		aomclem6.c	. . 3 ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))
3		aomclem6.d	. . 3 ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
4		aomclem6.e	. . 3 ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}
5		aomclem6.f	. . 3 ⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
6		aomclem6.g	. . 3 ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
7		aomclem6.h	. . 3 ⊢ 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))
8		aomclem6.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
9		aomclem6.y	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	aomclem6 43216	. 2 ⊢ (𝜑 → (𝐻‘𝐴) We (𝑅1‘𝐴))
11		fvex 6844	. . 3 ⊢ (𝐻‘𝐴) ∈ V
12		weeq1 5608	. . 3 ⊢ (𝑏 = (𝐻‘𝐴) → (𝑏 We (𝑅1‘𝐴) ↔ (𝐻‘𝐴) We (𝑅1‘𝐴)))
13	11, 12	spcev 3557	. 2 ⊢ ((𝐻‘𝐴) We (𝑅1‘𝐴) → ∃𝑏 𝑏 We (𝑅1‘𝐴))
14	10, 13	syl 17	1 ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ∃wrex 3057  Vcvv 3437   ∖ cdif 3895   ∩ cin 3897  ∅c0 4282  ifcif 4476  𝒫 cpw 4551  {csn 4577  ∪ cuni 4860  ∩ cint 4899   class class class wbr 5095  {copab 5157   ↦ cmpt 5176   E cep 5520   We wwe 5573   × cxp 5619  ^◡ccnv 5620  dom cdm 5621  ran crn 5622   “ cima 5624  Oncon0 6314  suc csuc 6316  ‘cfv 6489  recscrecs 8299  Fincfn 8879  supcsup 9335  𝑅₁cr1 9666  rankcrnk 9667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-map 8761  df-en 8880  df-fin 8883  df-sup 9337  df-r1 9668  df-rank 9669

This theorem is referenced by:  aomclem8  43218