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Theorem isf32lem11 9836
 Description: Lemma for isfin3-2 9840. Remove hypotheses from isf32lem10 9835. (Contributed by Stefan O'Rear, 17-May-2015.)
Assertion
Ref Expression
isf32lem11 ((𝐺𝑉 ∧ (𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ 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 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1133 . . 3 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → 𝐹:ω⟶𝒫 𝐺)
2 suceq 6239 . . . . . . . 8 (𝑏 = 𝑐 → suc 𝑏 = suc 𝑐)
32fveq2d 6667 . . . . . . 7 (𝑏 = 𝑐 → (𝐹‘suc 𝑏) = (𝐹‘suc 𝑐))
4 fveq2 6663 . . . . . . 7 (𝑏 = 𝑐 → (𝐹𝑏) = (𝐹𝑐))
53, 4sseq12d 3927 . . . . . 6 (𝑏 = 𝑐 → ((𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ↔ (𝐹‘suc 𝑐) ⊆ (𝐹𝑐)))
65cbvralvw 3361 . . . . 5 (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ↔ ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹𝑐))
76biimpi 219 . . . 4 (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) → ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹𝑐))
873ad2ant2 1131 . . 3 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹𝑐))
9 simp3 1135 . . 3 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → ¬ ran 𝐹 ∈ ran 𝐹)
10 suceq 6239 . . . . . 6 (𝑒 = 𝑑 → suc 𝑒 = suc 𝑑)
1110fveq2d 6667 . . . . 5 (𝑒 = 𝑑 → (𝐹‘suc 𝑒) = (𝐹‘suc 𝑑))
12 fveq2 6663 . . . . 5 (𝑒 = 𝑑 → (𝐹𝑒) = (𝐹𝑑))
1311, 12psseq12d 4002 . . . 4 (𝑒 = 𝑑 → ((𝐹‘suc 𝑒) ⊊ (𝐹𝑒) ↔ (𝐹‘suc 𝑑) ⊊ (𝐹𝑑)))
1413cbvrabv 3404 . . 3 {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} = {𝑑 ∈ ω ∣ (𝐹‘suc 𝑑) ⊊ (𝐹𝑑)}
15 eqid 2758 . . 3 (𝑓 ∈ ω ↦ (𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)}) ≈ 𝑓)) = (𝑓 ∈ ω ↦ (𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)}) ≈ 𝑓))
16 eqid 2758 . . 3 (( ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} ↦ ((𝐹) ∖ (𝐹‘suc ))) ∘ (𝑓 ∈ ω ↦ (𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)}) ≈ 𝑓))) = (( ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} ↦ ((𝐹) ∖ (𝐹‘suc ))) ∘ (𝑓 ∈ ω ↦ (𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)}) ≈ 𝑓)))
17 eqid 2758 . . 3 (𝑘𝐺 ↦ (℩𝑙(𝑙 ∈ ω ∧ 𝑘 ∈ ((( ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} ↦ ((𝐹) ∖ (𝐹‘suc ))) ∘ (𝑓 ∈ ω ↦ (𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)}) ≈ 𝑓)))‘𝑙)))) = (𝑘𝐺 ↦ (℩𝑙(𝑙 ∈ ω ∧ 𝑘 ∈ ((( ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} ↦ ((𝐹) ∖ (𝐹‘suc ))) ∘ (𝑓 ∈ ω ↦ (𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)}) ≈ 𝑓)))‘𝑙))))
181, 8, 9, 14, 15, 16, 17isf32lem10 9835 . 2 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → (𝐺𝑉 → ω ≼* 𝐺))
1918impcom 411 1 ((𝐺𝑉 ∧ (𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹)) → ω ≼* 𝐺)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111  ∀wral 3070  {crab 3074   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860   ⊊ wpss 3861  𝒫 cpw 4497  ∩ cint 4841   class class class wbr 5036   ↦ cmpt 5116  ran crn 5529   ∘ ccom 5532  suc csuc 6176  ℩cio 6297  ⟶wf 6336  ‘cfv 6340  ℩crio 7113  ωcom 7585   ≈ cen 8537   ≼* cwdom 9074 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-isom 6349  df-riota 7114  df-om 7586  df-wrecs 7963  df-recs 8024  df-1o 8118  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-wdom 9075  df-card 9414 This theorem is referenced by:  isf32lem12  9837  fin33i  9842
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