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Theorem isf32lem11 9068
 Description: Lemma for isfin3-2 9072. Remove hypotheses from isf32lem10 9067. (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 1054 . . 3 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → 𝐹:ω⟶𝒫 𝐺)
2 suceq 5707 . . . . . . . 8 (𝑏 = 𝑐 → suc 𝑏 = suc 𝑐)
32fveq2d 6107 . . . . . . 7 (𝑏 = 𝑐 → (𝐹‘suc 𝑏) = (𝐹‘suc 𝑐))
4 fveq2 6103 . . . . . . 7 (𝑏 = 𝑐 → (𝐹𝑏) = (𝐹𝑐))
53, 4sseq12d 3597 . . . . . 6 (𝑏 = 𝑐 → ((𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ↔ (𝐹‘suc 𝑐) ⊆ (𝐹𝑐)))
65cbvralv 3147 . . . . 5 (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ↔ ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹𝑐))
76biimpi 205 . . . 4 (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) → ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹𝑐))
873ad2ant2 1076 . . 3 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹𝑐))
9 simp3 1056 . . 3 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → ¬ ran 𝐹 ∈ ran 𝐹)
10 suceq 5707 . . . . . 6 (𝑒 = 𝑑 → suc 𝑒 = suc 𝑑)
1110fveq2d 6107 . . . . 5 (𝑒 = 𝑑 → (𝐹‘suc 𝑒) = (𝐹‘suc 𝑑))
12 fveq2 6103 . . . . 5 (𝑒 = 𝑑 → (𝐹𝑒) = (𝐹𝑑))
1311, 12psseq12d 3663 . . . 4 (𝑒 = 𝑑 → ((𝐹‘suc 𝑒) ⊊ (𝐹𝑒) ↔ (𝐹‘suc 𝑑) ⊊ (𝐹𝑑)))
1413cbvrabv 3172 . . 3 {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} = {𝑑 ∈ ω ∣ (𝐹‘suc 𝑑) ⊊ (𝐹𝑑)}
15 eqid 2610 . . 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 2610 . . 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 2610 . . 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 9067 . 2 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → (𝐺𝑉 → ω ≼* 𝐺))
1918impcom 445 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 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540   ⊊ wpss 3541  𝒫 cpw 4108  ∩ cint 4410   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039   ∘ ccom 5042  suc csuc 5642  ℩cio 5766  ⟶wf 5800  ‘cfv 5804  ℩crio 6510  ωcom 6957   ≈ cen 7838   ≼* cwdom 8345 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-wdom 8347  df-card 8648 This theorem is referenced by:  isf32lem12  9069  fin33i  9074
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