Theorem ttukey2g 10272

Description: The Teichmüller-Tukey Lemma ttukey 10274 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 4066	. . . 4 ⊢ (∪ 𝐴 ∖ 𝐵) ⊆ ∪ 𝐴
2		ssnum 9795	. . . 4 ⊢ ((∪ 𝐴 ∈ dom card ∧ (∪ 𝐴 ∖ 𝐵) ⊆ ∪ 𝐴) → (∪ 𝐴 ∖ 𝐵) ∈ dom card)
3	1, 2	mpan2 688	. . 3 ⊢ (∪ 𝐴 ∈ dom card → (∪ 𝐴 ∖ 𝐵) ∈ dom card)
4		isnum3 9712	. . . . 5 ⊢ ((∪ 𝐴 ∖ 𝐵) ∈ dom card ↔ (card‘(∪ 𝐴 ∖ 𝐵)) ≈ (∪ 𝐴 ∖ 𝐵))
5		bren 8743	. . . . 5 ⊢ ((card‘(∪ 𝐴 ∖ 𝐵)) ≈ (∪ 𝐴 ∖ 𝐵) ↔ ∃𝑓 𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
6	4, 5	bitri 274	. . . 4 ⊢ ((∪ 𝐴 ∖ 𝐵) ∈ dom card ↔ ∃𝑓 𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
7		simp1 1135	. . . . . . 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 1136	. . . . . . 7 ⊢ ((𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝐵 ∈ 𝐴)
9		simp3 1137	. . . . . . 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 5812	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
11	10	unieqd 4853	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ∪ dom 𝑤 = ∪ dom 𝑧)
12	10, 11	eqeq12d 2754	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (dom 𝑤 = ∪ dom 𝑤 ↔ dom 𝑧 = ∪ dom 𝑧))
13	10	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (dom 𝑤 = ∅ ↔ dom 𝑧 = ∅))
14		rneq 5845	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → ran 𝑤 = ran 𝑧)
15	14	unieqd 4853	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ∪ ran 𝑤 = ∪ ran 𝑧)
16	13, 15	ifbieq2d 4485	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → if(dom 𝑤 = ∅, 𝐵, ∪ ran 𝑤) = if(dom 𝑧 = ∅, 𝐵, ∪ ran 𝑧))
17		id 22	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → 𝑤 = 𝑧)
18	17, 11	fveq12d 6781	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑤‘∪ dom 𝑤) = (𝑧‘∪ dom 𝑧))
19	11	fveq2d 6778	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (𝑓‘∪ dom 𝑤) = (𝑓‘∪ dom 𝑧))
20	19	sneqd 4573	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → {(𝑓‘∪ dom 𝑤)} = {(𝑓‘∪ dom 𝑧)})
21	18, 20	uneq12d 4098	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → ((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) = ((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}))
22	21	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴 ↔ ((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴))
23	22, 20	ifbieq1d 4483	. . . . . . . . . . 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 4098	. . . . . . . . . 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 4487	. . . . . . . . 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 5187	. . . . . . . 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 8205	. . . . . . . 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 10271	. . . . . 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 1121	. . . . 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 1933	. . . 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 216	. . 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 17	. 2 ⊢ (∪ 𝐴 ∈ dom card → ((𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)))
34	33	3impib 1115	1 ⊢ ((∪ 𝐴 ∈ dom card ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887   ⊊ wpss 3888  ∅c0 4256  ifcif 4459  𝒫 cpw 4533  {csn 4561  ∪ cuni 4839   class class class wbr 5074   ↦ cmpt 5157  dom cdm 5589  ran crn 5590  –1-1-onto→wf1o 6432  ‘cfv 6433  recscrecs 8201   ≈ cen 8730  Fincfn 8733  cardccrd 9693

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-fin 8737  df-card 9697

This theorem is referenced by:  ttukeyg  10273