Theorem ttukey2g 10407

Description: The Teichmüller-Tukey Lemma ttukey 10409 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 4086	. . . 4 ⊢ (∪ 𝐴 ∖ 𝐵) ⊆ ∪ 𝐴
2		ssnum 9930	. . . 4 ⊢ ((∪ 𝐴 ∈ dom card ∧ (∪ 𝐴 ∖ 𝐵) ⊆ ∪ 𝐴) → (∪ 𝐴 ∖ 𝐵) ∈ dom card)
3	1, 2	mpan2 691	. . 3 ⊢ (∪ 𝐴 ∈ dom card → (∪ 𝐴 ∖ 𝐵) ∈ dom card)
4		isnum3 9847	. . . . 5 ⊢ ((∪ 𝐴 ∖ 𝐵) ∈ dom card ↔ (card‘(∪ 𝐴 ∖ 𝐵)) ≈ (∪ 𝐴 ∖ 𝐵))
5		bren 8879	. . . . 5 ⊢ ((card‘(∪ 𝐴 ∖ 𝐵)) ≈ (∪ 𝐴 ∖ 𝐵) ↔ ∃𝑓 𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
6	4, 5	bitri 275	. . . 4 ⊢ ((∪ 𝐴 ∖ 𝐵) ∈ dom card ↔ ∃𝑓 𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
7		simp1 1136	. . . . . . 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 1137	. . . . . . 7 ⊢ ((𝑓:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝐵 ∈ 𝐴)
9		simp3 1138	. . . . . . 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 5843	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
11	10	unieqd 4872	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ∪ dom 𝑤 = ∪ dom 𝑧)
12	10, 11	eqeq12d 2747	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (dom 𝑤 = ∪ dom 𝑤 ↔ dom 𝑧 = ∪ dom 𝑧))
13	10	eqeq1d 2733	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (dom 𝑤 = ∅ ↔ dom 𝑧 = ∅))
14		rneq 5876	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → ran 𝑤 = ran 𝑧)
15	14	unieqd 4872	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ∪ ran 𝑤 = ∪ ran 𝑧)
16	13, 15	ifbieq2d 4502	. . . . . . . . . 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 6829	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑤‘∪ dom 𝑤) = (𝑧‘∪ dom 𝑧))
19	11	fveq2d 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (𝑓‘∪ dom 𝑤) = (𝑓‘∪ dom 𝑧))
20	19	sneqd 4588	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → {(𝑓‘∪ dom 𝑤)} = {(𝑓‘∪ dom 𝑧)})
21	18, 20	uneq12d 4119	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → ((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) = ((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}))
22	21	eleq1d 2816	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴 ↔ ((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴))
23	22, 20	ifbieq1d 4500	. . . . . . . . . . 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 4119	. . . . . . . . . 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 4504	. . . . . . . . 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 5195	. . . . . . . 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 8293	. . . . . . . 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 10406	. . . . . 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 1122	. . . . 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 1931	. . . 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 217	. . 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 1116	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 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4283  ifcif 4475  𝒫 cpw 4550  {csn 4576  ∪ cuni 4859   class class class wbr 5091   ↦ cmpt 5172  dom cdm 5616  ran crn 5617  –1-1-onto→wf1o 6480  ‘cfv 6481  recscrecs 8290   ≈ cen 8866  Fincfn 8869  cardccrd 9828

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-fin 8873  df-card 9832

This theorem is referenced by:  ttukeyg  10408