Theorem aomclem5 38587

Description: Lemma for dfac11 38591. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Hypotheses

Ref	Expression
aomclem5.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)))}
aomclem5.c	⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))
aomclem5.d	⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
aomclem5.e	⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}
aomclem5.f	⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
aomclem5.g	⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
aomclem5.on	⊢ (𝜑 → dom 𝑧 ∈ On)
aomclem5.we	⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
aomclem5.a	⊢ (𝜑 → 𝐴 ∈ On)
aomclem5.za	⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)
aomclem5.y	⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))

Assertion

Ref	Expression
aomclem5	⊢ (𝜑 → 𝐺 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)   𝐹(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐺(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem aomclem5

Step	Hyp	Ref	Expression
1		aomclem5.f	. . . . . 6 ⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
2		aomclem5.on	. . . . . . 7 ⊢ (𝜑 → dom 𝑧 ∈ On)
3	2	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 ∈ On)
4		simpr 479	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 = ∪ dom 𝑧)
5		aomclem5.we	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
6	5	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
7	1, 3, 4, 6	aomclem4 38586	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → 𝐹 We (𝑅1‘dom 𝑧))
8		iftrue 4313	. . . . . . 7 ⊢ (dom 𝑧 = ∪ dom 𝑧 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐹)
9	8	adantl 475	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐹)
10		eqidd 2779	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
11	9, 10	weeq12d 38569	. . . . 5 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐹 We (𝑅1‘dom 𝑧)))
12	7, 11	mpbird 249	. . . 4 ⊢ ((𝜑 ∧ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
13		aomclem5.b	. . . . . 6 ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}
14		aomclem5.c	. . . . . 6 ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))
15		aomclem5.d	. . . . . 6 ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
16		aomclem5.e	. . . . . 6 ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}
17	2	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 ∈ On)
18		eloni 5986	. . . . . . . 8 ⊢ (dom 𝑧 ∈ On → Ord dom 𝑧)
19		orduniorsuc 7308	. . . . . . . 8 ⊢ (Ord dom 𝑧 → (dom 𝑧 = ∪ dom 𝑧 ∨ dom 𝑧 = suc ∪ dom 𝑧))
20	2, 18, 19	3syl 18	. . . . . . 7 ⊢ (𝜑 → (dom 𝑧 = ∪ dom 𝑧 ∨ dom 𝑧 = suc ∪ dom 𝑧))
21	20	orcanai 988	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 = suc ∪ dom 𝑧)
22	5	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
23		aomclem5.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
24	23	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → 𝐴 ∈ On)
25		aomclem5.za	. . . . . . 7 ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)
26	25	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → dom 𝑧 ⊆ 𝐴)
27		aomclem5.y	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
28	27	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
29	13, 14, 15, 16, 17, 21, 22, 24, 26, 28	aomclem3 38585	. . . . 5 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → 𝐸 We (𝑅1‘dom 𝑧))
30		iffalse 4316	. . . . . . 7 ⊢ (¬ dom 𝑧 = ∪ dom 𝑧 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐸)
31	30	adantl 475	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) = 𝐸)
32		eqidd 2779	. . . . . 6 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
33	31, 32	weeq12d 38569	. . . . 5 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐸 We (𝑅1‘dom 𝑧)))
34	29, 33	mpbird 249	. . . 4 ⊢ ((𝜑 ∧ ¬ dom 𝑧 = ∪ dom 𝑧) → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
35	12, 34	pm2.61dan 803	. . 3 ⊢ (𝜑 → if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
36		weinxp 5434	. . 3 ⊢ (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧))
37	35, 36	sylib 210	. 2 ⊢ (𝜑 → (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧))
38		aomclem5.g	. . 3 ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
39		weeq1 5343	. . 3 ⊢ (𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) → (𝐺 We (𝑅1‘dom 𝑧) ↔ (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧)))
40	38, 39	ax-mp 5	. 2 ⊢ (𝐺 We (𝑅1‘dom 𝑧) ↔ (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧))
41	37, 40	sylibr 226	1 ⊢ (𝜑 → 𝐺 We (𝑅1‘dom 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  ∃wrex 3091  Vcvv 3398   ∖ cdif 3789   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  ifcif 4307  𝒫 cpw 4379  {csn 4398  ∪ cuni 4671  ∩ cint 4710   class class class wbr 4886  {copab 4948   ↦ cmpt 4965   E cep 5265   We wwe 5313   × cxp 5353  ^◡ccnv 5354  dom cdm 5355  ran crn 5356   “ cima 5358  Ord word 5975  Oncon0 5976  suc csuc 5978  ‘cfv 6135  recscrecs 7750  Fincfn 8241  supcsup 8634  𝑅₁cr1 8922  rankcrnk 8923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-er 8026  df-map 8142  df-en 8242  df-fin 8245  df-sup 8636  df-r1 8924  df-rank 8925

This theorem is referenced by:  aomclem6  38588