Theorem isf32lem11 10386

Description: Lemma for isfin3-2 10390. Remove hypotheses from isf32lem10 10385. (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 6431	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → suc 𝑏 = suc 𝑐)
3	2	fveq2d 6891	. . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝐹‘suc 𝑏) = (𝐹‘suc 𝑐))
4		fveq2 6887	. . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝐹‘𝑏) = (𝐹‘𝑐))
5	3, 4	sseq12d 3999	. . . . . 6 ⊢ (𝑏 = 𝑐 → ((𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ↔ (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐)))
6	5	cbvralvw 3224	. . . . 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 6431	. . . . . 6 ⊢ (𝑒 = 𝑑 → suc 𝑒 = suc 𝑑)
11	10	fveq2d 6891	. . . . 5 ⊢ (𝑒 = 𝑑 → (𝐹‘suc 𝑒) = (𝐹‘suc 𝑑))
12		fveq2 6887	. . . . 5 ⊢ (𝑒 = 𝑑 → (𝐹‘𝑒) = (𝐹‘𝑑))
13	11, 12	psseq12d 4079	. . . 4 ⊢ (𝑒 = 𝑑 → ((𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒) ↔ (𝐹‘suc 𝑑) ⊊ (𝐹‘𝑑)))
14	13	cbvrabv 3431	. . 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 10385	. 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 2107  ∀wral 3050  {crab 3420   ∖ cdif 3930   ∩ cin 3932   ⊆ wss 3933   ⊊ wpss 3934  𝒫 cpw 4582  ∩ cint 4928   class class class wbr 5125   ↦ cmpt 5207  ran crn 5668   ∘ ccom 5671  suc csuc 6367  ℩cio 6493  ⟶wf 6538  ‘cfv 6542  ℩crio 7370  ωcom 7870   ≈ cen 8965   ≼^* cwdom 9587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-int 4929  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7371  df-ov 7417  df-om 7871  df-2nd 7998  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-1o 8489  df-er 8728  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-wdom 9588  df-card 9962

This theorem is referenced by:  isf32lem12  10387  fin33i  10392