Theorem fin23lem33 10340

Description: Lemma for fin23 10384. Discharge hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Hypothesis

Ref	Expression
fin23lem33.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}

Assertion

Ref	Expression
fin23lem33	⊢ (𝐺 ∈ 𝐹 → ∃𝑓∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏)))

Distinct variable groups:   𝑎,𝑏,𝑓,𝑔,𝑥,𝐺   𝐹,𝑎

Allowed substitution hints:   𝐹(𝑥,𝑓,𝑔,𝑏)

Proof of Theorem fin23lem33

Dummy variables 𝑐 𝑑 𝑒 𝑖 𝑗 𝑘 𝑙 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6892	. . . . . . 7 ⊢ (𝑗 = 𝑐 → (𝑒‘𝑗) = (𝑒‘𝑐))
2	1	ineq1d 4212	. . . . . 6 ⊢ (𝑗 = 𝑐 → ((𝑒‘𝑗) ∩ 𝑘) = ((𝑒‘𝑐) ∩ 𝑘))
3	2	eqeq1d 2735	. . . . 5 ⊢ (𝑗 = 𝑐 → (((𝑒‘𝑗) ∩ 𝑘) = ∅ ↔ ((𝑒‘𝑐) ∩ 𝑘) = ∅))
4	3, 2	ifbieq2d 4555	. . . 4 ⊢ (𝑗 = 𝑐 → if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘)) = if(((𝑒‘𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑐) ∩ 𝑘)))
5		ineq2 4207	. . . . . 6 ⊢ (𝑘 = 𝑑 → ((𝑒‘𝑐) ∩ 𝑘) = ((𝑒‘𝑐) ∩ 𝑑))
6	5	eqeq1d 2735	. . . . 5 ⊢ (𝑘 = 𝑑 → (((𝑒‘𝑐) ∩ 𝑘) = ∅ ↔ ((𝑒‘𝑐) ∩ 𝑑) = ∅))
7		id 22	. . . . 5 ⊢ (𝑘 = 𝑑 → 𝑘 = 𝑑)
8	6, 7, 5	ifbieq12d 4557	. . . 4 ⊢ (𝑘 = 𝑑 → if(((𝑒‘𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑐) ∩ 𝑘)) = if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑)))
9	4, 8	cbvmpov 7504	. . 3 ⊢ (𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑)))
10		eqid 2733	. . 3 ⊢ ∪ ran 𝑒 = ∪ ran 𝑒
11		seqomeq12 8454	. . 3 ⊢ (((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑))) ∧ ∪ ran 𝑒 = ∪ ran 𝑒) → seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) = seqω((𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑))), ∪ ran 𝑒))
12	9, 10, 11	mp2an 691	. 2 ⊢ seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) = seqω((𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒‘𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒‘𝑐) ∩ 𝑑))), ∪ ran 𝑒)
13		fin23lem33.f	. 2 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
14		fveq2 6892	. . . 4 ⊢ (𝑙 = 𝑦 → (𝑒‘𝑙) = (𝑒‘𝑦))
15	14	sseq2d 4015	. . 3 ⊢ (𝑙 = 𝑦 → (∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙) ↔ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑦)))
16	15	cbvrabv 3443	. 2 ⊢ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} = {𝑦 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑦)}
17		eqid 2733	. 2 ⊢ (𝑔 ∈ ω ↦ (℩𝑥 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)}) ≈ 𝑔)) = (𝑔 ∈ ω ↦ (℩𝑥 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)}) ≈ 𝑔))
18		eqid 2733	. 2 ⊢ (𝑔 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})) ≈ 𝑔)) = (𝑔 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})) ≈ 𝑔))
19		eqid 2733	. 2 ⊢ if({𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} ∈ Fin, (𝑒 ∘ (𝑔 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})) ≈ 𝑔))), ((𝑖 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} ↦ ((𝑒‘𝑖) ∖ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒))) ∘ (𝑔 ∈ ω ↦ (℩𝑥 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)}) ≈ 𝑔)))) = if({𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} ∈ Fin, (𝑒 ∘ (𝑔 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)})) ≈ 𝑔))), ((𝑖 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} ↦ ((𝑒‘𝑖) ∖ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒))) ∘ (𝑔 ∈ ω ↦ (℩𝑥 ∈ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ∩ ran seqω((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒‘𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒‘𝑗) ∩ 𝑘))), ∪ ran 𝑒) ⊆ (𝑒‘𝑙)}) ≈ 𝑔))))
20	12, 13, 16, 17, 18, 19	fin23lem32 10339	1 ⊢ (𝐺 ∈ 𝐹 → ∃𝑓∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏)))

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1540   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710  ∀wral 3062  {crab 3433  Vcvv 3475   ∖ cdif 3946   ∩ cin 3948   ⊆ wss 3949   ⊊ wpss 3950  ∅c0 4323  ifcif 4529  𝒫 cpw 4603  ∪ cuni 4909  ∩ cint 4951   class class class wbr 5149   ↦ cmpt 5232  ran crn 5678   ∘ ccom 5681  suc csuc 6367  –1-1→wf1 6541  ‘cfv 6544  ℩crio 7364  (class class class)co 7409   ∈ cmpo 7411  ωcom 7855  seq_ωcseqom 8447   ↑_m cmap 8820   ≈ cen 8936  Fincfn 8939

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-seqom 8448  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934

This theorem is referenced by:  fin23lem41  10347