Theorem isf32lem11 10271

Description: Lemma for isfin3-2 10275. Remove hypotheses from isf32lem10 10270. (Contributed by Stefan O'Rear, 17-May-2015.)

Assertion

Ref	Expression
isf32lem11	⊢ ((𝐺 ∈ 𝑉 ∧ (𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹)) → ω ≼* 𝐺)

Distinct variable groups:   𝐹,𝑏   𝐺,𝑏

Allowed substitution hint:   𝑉(𝑏)

Proof of Theorem isf32lem11

Dummy variables 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . 3 ⊢ ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹) → 𝐹:ω⟶𝒫 𝐺)
2		suceq 6383	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → suc 𝑏 = suc 𝑐)
3	2	fveq2d 6836	. . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝐹‘suc 𝑏) = (𝐹‘suc 𝑐))
4		fveq2 6832	. . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝐹‘𝑏) = (𝐹‘𝑐))
5	3, 4	sseq12d 3965	. . . . . 6 ⊢ (𝑏 = 𝑐 → ((𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ↔ (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐)))
6	5	cbvralvw 3212	. . . . 5 ⊢ (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ↔ ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐))
7	6	biimpi 216	. . . 4 ⊢ (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) → ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐))
8	7	3ad2ant2 1134	. . 3 ⊢ ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹) → ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐))
9		simp3 1138	. . 3 ⊢ ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹) → ¬ ∩ ran 𝐹 ∈ ran 𝐹)
10		suceq 6383	. . . . . 6 ⊢ (𝑒 = 𝑑 → suc 𝑒 = suc 𝑑)
11	10	fveq2d 6836	. . . . 5 ⊢ (𝑒 = 𝑑 → (𝐹‘suc 𝑒) = (𝐹‘suc 𝑑))
12		fveq2 6832	. . . . 5 ⊢ (𝑒 = 𝑑 → (𝐹‘𝑒) = (𝐹‘𝑑))
13	11, 12	psseq12d 4047	. . . 4 ⊢ (𝑒 = 𝑑 → ((𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒) ↔ (𝐹‘suc 𝑑) ⊊ (𝐹‘𝑑)))
14	13	cbvrabv 3407	. . 3 ⊢ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} = {𝑑 ∈ ω ∣ (𝐹‘suc 𝑑) ⊊ (𝐹‘𝑑)}
15		eqid 2734	. . 3 ⊢ (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓)) = (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓))
16		eqid 2734	. . 3 ⊢ ((ℎ ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} ↦ ((𝐹‘ℎ) ∖ (𝐹‘suc ℎ))) ∘ (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓))) = ((ℎ ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} ↦ ((𝐹‘ℎ) ∖ (𝐹‘suc ℎ))) ∘ (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓)))
17		eqid 2734	. . 3 ⊢ (𝑘 ∈ 𝐺 ↦ (℩𝑙(𝑙 ∈ ω ∧ 𝑘 ∈ (((ℎ ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} ↦ ((𝐹‘ℎ) ∖ (𝐹‘suc ℎ))) ∘ (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓)))‘𝑙)))) = (𝑘 ∈ 𝐺 ↦ (℩𝑙(𝑙 ∈ ω ∧ 𝑘 ∈ (((ℎ ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} ↦ ((𝐹‘ℎ) ∖ (𝐹‘suc ℎ))) ∘ (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓)))‘𝑙))))
18	1, 8, 9, 14, 15, 16, 17	isf32lem10 10270	. 2 ⊢ ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹) → (𝐺 ∈ 𝑉 → ω ≼* 𝐺))
19	18	impcom 407	1 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹)) → ω ≼* 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2113  ∀wral 3049  {crab 3397   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899   ⊊ wpss 3900  𝒫 cpw 4552  ∩ cint 4900   class class class wbr 5096   ↦ cmpt 5177  ran crn 5623   ∘ ccom 5626  suc csuc 6317  ℩cio 6444  ⟶wf 6486  ‘cfv 6490  ℩crio 7312  ωcom 7806   ≈ cen 8878   ≼^* cwdom 9467

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-wdom 9468  df-card 9849

This theorem is referenced by:  isf32lem12  10272  fin33i  10277