Theorem ttukey2g 10518

Description: The Teichmüller-Tukey Lemma ttukey 10520 with a slightly stronger conclusion: we can set up the maximal element of 𝐴 so that it also contains some given 𝐵 ∈ 𝐴 as a subset. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
ttukey2g	⊢ ((∪ 𝐴 ∈ dom card ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem ttukey2g

Dummy variables 𝑤 𝑓 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difss 4093	. . . 4 ⊢ (∪ 𝐴 ∖ 𝐵) ⊆ ∪ 𝐴
2		ssnum 10042	. . . 4 ⊢ ((∪ 𝐴 ∈ dom card ∧ (∪ 𝐴 ∖ 𝐵) ⊆ ∪ 𝐴) → (∪ 𝐴 ∖ 𝐵) ∈ dom card)
3	1, 2	mpan2 704	. . 3 ⊢ (∪ 𝐴 ∈ dom card → (∪ 𝐴 ∖ 𝐵) ∈ dom card)
4		isnum3 9959	. . . . 5 ⊢ ((∪ 𝐴 ∖ 𝐵) ∈ dom card ↔ (card‘(∪ 𝐴 ∖ 𝐵)) ≈ (∪ 𝐴 ∖ 𝐵))
5		bren 8962	. . . . 5 ⊢ ((card‘(∪ 𝐴 ∖ 𝐵)) ≈ (∪ 𝐴 ∖ 𝐵) ↔ ∃𝑓 𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
6	4, 5	bitri 278	. . . 4 ⊢ ((∪ 𝐴 ∖ 𝐵) ∈ dom card ↔ ∃𝑓 𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
7		simp1 1154	. . . . . . 7 ⊢ ((𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
8		simp2 1155	. . . . . . 7 ⊢ ((𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝐵 ∈ 𝐴)
9		simp3 1156	. . . . . . 7 ⊢ ((𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
10		dmeq 5898	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
11	10	unieqd 4890	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ∪ dom 𝑤 = ∪ dom 𝑧)
12	10, 11	eqeq12d 2782	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (dom 𝑤 = ∪ dom 𝑤 ↔ dom 𝑧 = ∪ dom 𝑧))
13	10	eqeq1d 2768	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (dom 𝑤 = ∅ ↔ dom 𝑧 = ∅))
14		rneq 5931	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → ran 𝑤 = ran 𝑧)
15	14	unieqd 4890	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ∪ ran 𝑤 = ∪ ran 𝑧)
16	13, 15	ifbieq2d 4519	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → if(dom 𝑤 = ∅, 𝐵, ∪ ran 𝑤) = if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧))
17		id 23	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → 𝑤 = 𝑧)
18	17, 11	fveq12d 6895	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑤‘∪ dom 𝑤) = (𝑧‘∪ dom 𝑧))
19	11	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (𝑓‘∪ dom 𝑤) = (𝑓‘∪ dom 𝑧))
20	19	sneqd 4606	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → {(𝑓‘∪ dom 𝑤)} = {(𝑓‘∪ dom 𝑧)})
21	18, 20	uneq12d 4126	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → ((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) = ((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}))
22	21	eleq1d 2851	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴 ↔ ((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴))
23	22, 20	ifbieq1d 4517	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → if(((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑤)}, ∅) = if(((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑧)}, ∅))
24	18, 23	uneq12d 4126	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → ((𝑤‘∪ dom 𝑤) ∪ if(((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑤)}, ∅)) = ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑧)}, ∅)))
25	12, 16, 24	ifbieq12d 4521	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → if(dom 𝑤 = ∪ dom 𝑤, if(dom 𝑤 = ∅, 𝐵, ∪ ran 𝑤), ((𝑤‘∪ dom 𝑤) ∪ if(((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑤)}, ∅))) = if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑧)}, ∅))))
26	25	cbvmptv 5220	. . . . . . . 8 ⊢ (𝑤 ∈ V ↦ if(dom 𝑤 = ∪ dom 𝑤, if(dom 𝑤 = ∅, 𝐵, ∪ ran 𝑤), ((𝑤‘∪ dom 𝑤) ∪ if(((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑤)}, ∅)))) = (𝑧 ∈ V ↦ if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑧)}, ∅))))
27		recseq 8369	. . . . . . . 8 ⊢ ((𝑤 ∈ V ↦ if(dom 𝑤 = ∪ dom 𝑤, if(dom 𝑤 = ∅, 𝐵, ∪ ran 𝑤), ((𝑤‘∪ dom 𝑤) ∪ if(((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑤)}, ∅)))) = (𝑧 ∈ V ↦ if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑧)}, ∅)))) → recs((𝑤 ∈ V ↦ if(dom 𝑤 = ∪ dom 𝑤, if(dom 𝑤 = ∅, 𝐵, ∪ ran 𝑤), ((𝑤‘∪ dom 𝑤) ∪ if(((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑤)}, ∅))))) = recs((𝑧 ∈ V ↦ if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑧)}, ∅))))))
28	26, 27	ax-mp 5	. . . . . . 7 ⊢ recs((𝑤 ∈ V ↦ if(dom 𝑤 = ∪ dom 𝑤, if(dom 𝑤 = ∅, 𝐵, ∪ ran 𝑤), ((𝑤‘∪ dom 𝑤) ∪ if(((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑤)}, ∅))))) = recs((𝑧 ∈ V ↦ if(dom 𝑧 = ∪ dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧), ((𝑧‘∪ dom 𝑧) ∪ if(((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴, {(𝑓‘∪ dom 𝑧)}, ∅)))))
29	7, 8, 9, 28	ttukeylem7 10517	. . . . . 6 ⊢ ((𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))
30	29	3expib 1140	. . . . 5 ⊢ (𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) → ((𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)))
31	30	exlimiv 1963	. . . 4 ⊢ (∃𝑓 𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) → ((𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)))
32	6, 31	sylbi 220	. . 3 ⊢ ((∪ 𝐴 ∖ 𝐵) ∈ dom card → ((𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)))
33	3, 32	syl 18	. 2 ⊢ (∪ 𝐴 ∈ dom card → ((𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)))
34	33	3impib 1134	1 ⊢ ((∪ 𝐴 ∈ dom card ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092  Vcvv 3458   ∖ cdif 3905   ∪ cun 3906   ∩ cin 3907   ⊆ wss 3908   ⊊ wpss 3909  ∅c0 4289  ifcif 4492  𝒫 cpw 4567  {csn 4594  ∪ cuni 4877   class class class wbr 5114   ↦ cmpt 5197  dom cdm 5666  ran crn 5667  –1-1-onto→wf1o 6542  ‘cfv 6543  recscrecs 8366   ≈ cen 8949  Fincfn 8952  cardccrd 9940

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-om 7872  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-1o 8462  df-er 8703  df-en 8953  df-dom 8954  df-fin 8956  df-card 9944

This theorem is used by:  ttukeyg  10519