Theorem isf32lem11 10432

Description: Lemma for isfin3-2 10436. Remove hypotheses from isf32lem10 10431. (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 6461	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → suc 𝑏 = suc 𝑐)
3	2	fveq2d 6924	. . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝐹‘suc 𝑏) = (𝐹‘suc 𝑐))
4		fveq2 6920	. . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝐹‘𝑏) = (𝐹‘𝑐))
5	3, 4	sseq12d 4042	. . . . . 6 ⊢ (𝑏 = 𝑐 → ((𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ↔ (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐)))
6	5	cbvralvw 3243	. . . . 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 6461	. . . . . 6 ⊢ (𝑒 = 𝑑 → suc 𝑒 = suc 𝑑)
11	10	fveq2d 6924	. . . . 5 ⊢ (𝑒 = 𝑑 → (𝐹‘suc 𝑒) = (𝐹‘suc 𝑑))
12		fveq2 6920	. . . . 5 ⊢ (𝑒 = 𝑑 → (𝐹‘𝑒) = (𝐹‘𝑑))
13	11, 12	psseq12d 4120	. . . 4 ⊢ (𝑒 = 𝑑 → ((𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒) ↔ (𝐹‘suc 𝑑) ⊊ (𝐹‘𝑑)))
14	13	cbvrabv 3454	. . 3 ⊢ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} = {𝑑 ∈ ω ∣ (𝐹‘suc 𝑑) ⊊ (𝐹‘𝑑)}
15		eqid 2740	. . 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 2740	. . 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 2740	. . 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 10431	. 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 1087   ∈ wcel 2108  ∀wral 3067  {crab 3443   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976   ⊊ wpss 3977  𝒫 cpw 4622  ∩ cint 4970   class class class wbr 5166   ↦ cmpt 5249  ran crn 5701   ∘ ccom 5704  suc csuc 6397  ℩cio 6523  ⟶wf 6569  ‘cfv 6573  ℩crio 7403  ωcom 7903   ≈ cen 9000   ≼^* cwdom 9633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-wdom 9634  df-card 10008

This theorem is referenced by:  isf32lem12  10433  fin33i  10438