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Theorem fin23lem29 10381
Description: Lemma for fin23 10429. The residual is built from the same elements as the previous sequence. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem.a 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
fin23lem17.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.b 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
fin23lem.c 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
fin23lem.d 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
Assertion
Ref Expression
fin23lem29 ran 𝑍 ran 𝑡
Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎
Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem29
StepHypRef Expression
1 fin23lem.e . 2 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
2 eqif 4567 . . 3 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
32biimpi 216 . 2 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) → ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
4 rneq 5947 . . . . . 6 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
54unieqd 4920 . . . . 5 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6 rncoss 5986 . . . . . 6 ran (𝑡𝑅) ⊆ ran 𝑡
76unissi 4916 . . . . 5 ran (𝑡𝑅) ⊆ ran 𝑡
85, 7eqsstrdi 4028 . . . 4 (𝑍 = (𝑡𝑅) → ran 𝑍 ran 𝑡)
98adantl 481 . . 3 ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) → ran 𝑍 ran 𝑡)
10 rneq 5947 . . . . . 6 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
1110unieqd 4920 . . . . 5 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
12 rncoss 5986 . . . . . . 7 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
1312unissi 4916 . . . . . 6 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
14 unissb 4939 . . . . . . 7 ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ⊆ ran 𝑡 ↔ ∀𝑎 ∈ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))𝑎 ran 𝑡)
15 abid 2718 . . . . . . . . 9 (𝑎 ∈ {𝑎 ∣ ∃𝑧𝑃 𝑎 = ((𝑡𝑧) ∖ ran 𝑈)} ↔ ∃𝑧𝑃 𝑎 = ((𝑡𝑧) ∖ ran 𝑈))
16 fvssunirn 6939 . . . . . . . . . . . . 13 (𝑡𝑧) ⊆ ran 𝑡
1716a1i 11 . . . . . . . . . . . 12 (𝑧𝑃 → (𝑡𝑧) ⊆ ran 𝑡)
1817ssdifssd 4147 . . . . . . . . . . 11 (𝑧𝑃 → ((𝑡𝑧) ∖ ran 𝑈) ⊆ ran 𝑡)
19 sseq1 4009 . . . . . . . . . . 11 (𝑎 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎 ran 𝑡 ↔ ((𝑡𝑧) ∖ ran 𝑈) ⊆ ran 𝑡))
2018, 19syl5ibrcom 247 . . . . . . . . . 10 (𝑧𝑃 → (𝑎 = ((𝑡𝑧) ∖ ran 𝑈) → 𝑎 ran 𝑡))
2120rexlimiv 3148 . . . . . . . . 9 (∃𝑧𝑃 𝑎 = ((𝑡𝑧) ∖ ran 𝑈) → 𝑎 ran 𝑡)
2215, 21sylbi 217 . . . . . . . 8 (𝑎 ∈ {𝑎 ∣ ∃𝑧𝑃 𝑎 = ((𝑡𝑧) ∖ ran 𝑈)} → 𝑎 ran 𝑡)
23 eqid 2737 . . . . . . . . 9 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
2423rnmpt 5968 . . . . . . . 8 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑎 ∣ ∃𝑧𝑃 𝑎 = ((𝑡𝑧) ∖ ran 𝑈)}
2522, 24eleq2s 2859 . . . . . . 7 (𝑎 ∈ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) → 𝑎 ran 𝑡)
2614, 25mprgbir 3068 . . . . . 6 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ⊆ ran 𝑡
2713, 26sstri 3993 . . . . 5 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran 𝑡
2811, 27eqsstrdi 4028 . . . 4 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 ran 𝑡)
2928adantl 481 . . 3 ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) → ran 𝑍 ran 𝑡)
309, 29jaoi 858 . 2 (((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))) → ran 𝑍 ran 𝑡)
311, 3, 30mp2b 10 1 ran 𝑍 ran 𝑡
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  cin 3950  wss 3951  c0 4333  ifcif 4525  𝒫 cpw 4600   cuni 4907   cint 4946   class class class wbr 5143  cmpt 5225  ran crn 5686  ccom 5689  suc csuc 6386  cfv 6561  crio 7387  (class class class)co 7431  cmpo 7433  ωcom 7887  seqωcseqom 8487  m cmap 8866  cen 8982  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fv 6569
This theorem is referenced by:  fin23lem31  10383
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