Theorem ttukey2g 9937

Description: The Teichmüller-Tukey Lemma ttukey 9939 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 4107	. . . 4 ⊢ (∪ 𝐴 ∖ 𝐵) ⊆ ∪ 𝐴
2		ssnum 9464	. . . 4 ⊢ ((∪ 𝐴 ∈ dom card ∧ (∪ 𝐴 ∖ 𝐵) ⊆ ∪ 𝐴) → (∪ 𝐴 ∖ 𝐵) ∈ dom card)
3	1, 2	mpan2 689	. . 3 ⊢ (∪ 𝐴 ∈ dom card → (∪ 𝐴 ∖ 𝐵) ∈ dom card)
4		isnum3 9382	. . . . 5 ⊢ ((∪ 𝐴 ∖ 𝐵) ∈ dom card ↔ (card‘(∪ 𝐴 ∖ 𝐵)) ≈ (∪ 𝐴 ∖ 𝐵))
5		bren 8517	. . . . 5 ⊢ ((card‘(∪ 𝐴 ∖ 𝐵)) ≈ (∪ 𝐴 ∖ 𝐵) ↔ ∃𝑓 𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
6	4, 5	bitri 277	. . . 4 ⊢ ((∪ 𝐴 ∖ 𝐵) ∈ dom card ↔ ∃𝑓 𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
7		simp1 1132	. . . . . . 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 1133	. . . . . . 7 ⊢ ((𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝐵 ∈ 𝐴)
9		simp3 1134	. . . . . . 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 5771	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
11	10	unieqd 4851	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ∪ dom 𝑤 = ∪ dom 𝑧)
12	10, 11	eqeq12d 2837	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (dom 𝑤 = ∪ dom 𝑤 ↔ dom 𝑧 = ∪ dom 𝑧))
13	10	eqeq1d 2823	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (dom 𝑤 = ∅ ↔ dom 𝑧 = ∅))
14		rneq 5805	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → ran 𝑤 = ran 𝑧)
15	14	unieqd 4851	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ∪ ran 𝑤 = ∪ ran 𝑧)
16	13, 15	ifbieq2d 4491	. . . . . . . . . 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 6676	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑤‘∪ dom 𝑤) = (𝑧‘∪ dom 𝑧))
19	11	fveq2d 6673	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (𝑓‘∪ dom 𝑤) = (𝑓‘∪ dom 𝑧))
20	19	sneqd 4578	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → {(𝑓‘∪ dom 𝑤)} = {(𝑓‘∪ dom 𝑧)})
21	18, 20	uneq12d 4139	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → ((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) = ((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}))
22	21	eleq1d 2897	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴 ↔ ((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴))
23	22, 20	ifbieq1d 4489	. . . . . . . . . . 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 4139	. . . . . . . . . 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 4493	. . . . . . . . 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 5168	. . . . . . . 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 8009	. . . . . . . 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 9936	. . . . . 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 1118	. . . . 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 1927	. . . 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 219	. . 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 1112	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 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935   ⊊ wpss 3936  ∅c0 4290  ifcif 4466  𝒫 cpw 4538  {csn 4566  ∪ cuni 4837   class class class wbr 5065   ↦ cmpt 5145  dom cdm 5554  ran crn 5555  –1-1-onto→wf1o 6353  ‘cfv 6354  recscrecs 8006   ≈ cen 8505  Fincfn 8508  cardccrd 9363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-om 7580  df-wrecs 7946  df-recs 8007  df-1o 8101  df-er 8288  df-en 8509  df-dom 8510  df-fin 8512  df-card 9367

This theorem is referenced by:  ttukeyg  9938