Theorem isf32lem11 9386

Description: Lemma for isfin3-2 9390. Remove hypotheses from isf32lem10 9385. (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 1129	. . 3 ⊢ ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹) → 𝐹:ω⟶𝒫 𝐺)
2		suceq 5933	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → suc 𝑏 = suc 𝑐)
3	2	fveq2d 6336	. . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝐹‘suc 𝑏) = (𝐹‘suc 𝑐))
4		fveq2 6332	. . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝐹‘𝑏) = (𝐹‘𝑐))
5	3, 4	sseq12d 3781	. . . . . 6 ⊢ (𝑏 = 𝑐 → ((𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ↔ (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐)))
6	5	cbvralv 3319	. . . . 5 ⊢ (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ↔ ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐))
7	6	biimpi 206	. . . 4 ⊢ (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) → ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐))
8	7	3ad2ant2 1127	. . 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 1131	. . 3 ⊢ ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹) → ¬ ∩ ran 𝐹 ∈ ran 𝐹)
10		suceq 5933	. . . . . 6 ⊢ (𝑒 = 𝑑 → suc 𝑒 = suc 𝑑)
11	10	fveq2d 6336	. . . . 5 ⊢ (𝑒 = 𝑑 → (𝐹‘suc 𝑒) = (𝐹‘suc 𝑑))
12		fveq2 6332	. . . . 5 ⊢ (𝑒 = 𝑑 → (𝐹‘𝑒) = (𝐹‘𝑑))
13	11, 12	psseq12d 3849	. . . 4 ⊢ (𝑒 = 𝑑 → ((𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒) ↔ (𝐹‘suc 𝑑) ⊊ (𝐹‘𝑑)))
14	13	cbvrabv 3348	. . 3 ⊢ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} = {𝑑 ∈ ω ∣ (𝐹‘suc 𝑑) ⊊ (𝐹‘𝑑)}
15		eqid 2770	. . 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 2770	. . 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 2770	. . 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 9385	. 2 ⊢ ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹) → (𝐺 ∈ 𝑉 → ω ≼* 𝐺))
19	18	impcom 394	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 382   ∧ w3a 1070   ∈ wcel 2144  ∀wral 3060  {crab 3064   ∖ cdif 3718   ∩ cin 3720   ⊆ wss 3721   ⊊ wpss 3722  𝒫 cpw 4295  ∩ cint 4609   class class class wbr 4784   ↦ cmpt 4861  ran crn 5250   ∘ ccom 5253  suc csuc 5868  ℩cio 5992  ⟶wf 6027  ‘cfv 6031  ℩crio 6752  ωcom 7211   ≈ cen 8105   ≼^* cwdom 8617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-om 7212  df-wrecs 7558  df-recs 7620  df-1o 7712  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-wdom 8619  df-card 8964

This theorem is referenced by:  isf32lem12  9387  fin33i  9392