Theorem ttukey2g 10501

Description: The Teichmüller-Tukey Lemma ttukey 10503 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 4091	. . . 4 ⊢ (∪ 𝐴 ∖ 𝐵) ⊆ ∪ 𝐴
2		ssnum 10024	. . . 4 ⊢ ((∪ 𝐴 ∈ dom card ∧ (∪ 𝐴 ∖ 𝐵) ⊆ ∪ 𝐴) → (∪ 𝐴 ∖ 𝐵) ∈ dom card)
3	1, 2	mpan2 703	. . 3 ⊢ (∪ 𝐴 ∈ dom card → (∪ 𝐴 ∖ 𝐵) ∈ dom card)
4		isnum3 9941	. . . . 5 ⊢ ((∪ 𝐴 ∖ 𝐵) ∈ dom card ↔ (card‘(∪ 𝐴 ∖ 𝐵)) ≈ (∪ 𝐴 ∖ 𝐵))
5		bren 8954	. . . . 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 5895	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
11	10	unieqd 4886	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ∪ dom 𝑤 = ∪ dom 𝑧)
12	10, 11	eqeq12d 2779	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (dom 𝑤 = ∪ dom 𝑤 ↔ dom 𝑧 = ∪ dom 𝑧))
13	10	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (dom 𝑤 = ∅ ↔ dom 𝑧 = ∅))
14		rneq 5928	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → ran 𝑤 = ran 𝑧)
15	14	unieqd 4886	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ∪ ran 𝑤 = ∪ ran 𝑧)
16	13, 15	ifbieq2d 4515	. . . . . . . . . 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 6890	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑤‘∪ dom 𝑤) = (𝑧‘∪ dom 𝑧))
19	11	fveq2d 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (𝑓‘∪ dom 𝑤) = (𝑓‘∪ dom 𝑧))
20	19	sneqd 4602	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → {(𝑓‘∪ dom 𝑤)} = {(𝑓‘∪ dom 𝑧)})
21	18, 20	uneq12d 4124	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → ((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) = ((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}))
22	21	eleq1d 2848	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (((𝑤‘∪ dom 𝑤) ∪ {(𝑓‘∪ dom 𝑤)}) ∈ 𝐴 ↔ ((𝑧‘∪ dom 𝑧) ∪ {(𝑓‘∪ dom 𝑧)}) ∈ 𝐴))
23	22, 20	ifbieq1d 4513	. . . . . . . . . . 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 4124	. . . . . . . . . 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 4517	. . . . . . . . 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 5216	. . . . . . . 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 8361	. . . . . . . 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 10500	. . . . . 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 1960	. . . 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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906   ⊊ wpss 3907  ∅c0 4287  ifcif 4488  𝒫 cpw 4563  {csn 4590  ∪ cuni 4873   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5663  ran crn 5664  –1-1-onto→wf1o 6537  ‘cfv 6538  recscrecs 8358   ≈ cen 8941  Fincfn 8944  cardccrd 9922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-1o 8454  df-er 8695  df-en 8945  df-dom 8946  df-fin 8948  df-card 9926

This theorem is referenced by:  ttukeyg  10502