MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isf32lem11 Structured version   Visualization version   GIF version

Theorem isf32lem11 9391
Description: Lemma for isfin3-2 9395. Remove hypotheses from isf32lem10 9390. (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 1130 . . 3 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → 𝐹:ω⟶𝒫 𝐺)
2 suceq 5932 . . . . . . . 8 (𝑏 = 𝑐 → suc 𝑏 = suc 𝑐)
32fveq2d 6337 . . . . . . 7 (𝑏 = 𝑐 → (𝐹‘suc 𝑏) = (𝐹‘suc 𝑐))
4 fveq2 6333 . . . . . . 7 (𝑏 = 𝑐 → (𝐹𝑏) = (𝐹𝑐))
53, 4sseq12d 3783 . . . . . 6 (𝑏 = 𝑐 → ((𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ↔ (𝐹‘suc 𝑐) ⊆ (𝐹𝑐)))
65cbvralv 3320 . . . . 5 (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ↔ ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹𝑐))
76biimpi 206 . . . 4 (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) → ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹𝑐))
873ad2ant2 1128 . . 3 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹𝑐))
9 simp3 1132 . . 3 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → ¬ ran 𝐹 ∈ ran 𝐹)
10 suceq 5932 . . . . . 6 (𝑒 = 𝑑 → suc 𝑒 = suc 𝑑)
1110fveq2d 6337 . . . . 5 (𝑒 = 𝑑 → (𝐹‘suc 𝑒) = (𝐹‘suc 𝑑))
12 fveq2 6333 . . . . 5 (𝑒 = 𝑑 → (𝐹𝑒) = (𝐹𝑑))
1311, 12psseq12d 3851 . . . 4 (𝑒 = 𝑑 → ((𝐹‘suc 𝑒) ⊊ (𝐹𝑒) ↔ (𝐹‘suc 𝑑) ⊊ (𝐹𝑑)))
1413cbvrabv 3349 . . 3 {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹𝑒)} = {𝑑 ∈ ω ∣ (𝐹‘suc 𝑑) ⊊ (𝐹𝑑)}
15 eqid 2771 . . 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 2771 . . 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 2771 . . 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 9390 . 2 ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹) → (𝐺𝑉 → ω ≼* 𝐺))
1918impcom 394 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 382  w3a 1071  wcel 2145  wral 3061  {crab 3065  cdif 3720  cin 3722  wss 3723  wpss 3724  𝒫 cpw 4298   cint 4612   class class class wbr 4787  cmpt 4864  ran crn 5251  ccom 5254  suc csuc 5867  cio 5991  wf 6026  cfv 6030  crio 6756  ωcom 7216  cen 8110  * cwdom 8622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6757  df-om 7217  df-wrecs 7563  df-recs 7625  df-1o 7717  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-wdom 8624  df-card 8969
This theorem is referenced by:  isf32lem12  9392  fin33i  9397
  Copyright terms: Public domain W3C validator