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Theorem fin23lem29 9761
Description: Lemma for fin23 9809. 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 4490 . . 3 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
32biimpi 219 . 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 5793 . . . . . 6 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
54unieqd 4838 . . . . 5 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6 rncoss 5830 . . . . . 6 ran (𝑡𝑅) ⊆ ran 𝑡
76unissi 4833 . . . . 5 ran (𝑡𝑅) ⊆ ran 𝑡
85, 7eqsstrdi 4007 . . . 4 (𝑍 = (𝑡𝑅) → ran 𝑍 ran 𝑡)
98adantl 485 . . 3 ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) → ran 𝑍 ran 𝑡)
10 rneq 5793 . . . . . 6 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
1110unieqd 4838 . . . . 5 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
12 rncoss 5830 . . . . . . 7 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
1312unissi 4833 . . . . . 6 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
14 unissb 4856 . . . . . . 7 ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ⊆ ran 𝑡 ↔ ∀𝑎 ∈ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))𝑎 ran 𝑡)
15 abid 2806 . . . . . . . . 9 (𝑎 ∈ {𝑎 ∣ ∃𝑧𝑃 𝑎 = ((𝑡𝑧) ∖ ran 𝑈)} ↔ ∃𝑧𝑃 𝑎 = ((𝑡𝑧) ∖ ran 𝑈))
16 fvssunirn 6690 . . . . . . . . . . . . 13 (𝑡𝑧) ⊆ ran 𝑡
1716a1i 11 . . . . . . . . . . . 12 (𝑧𝑃 → (𝑡𝑧) ⊆ ran 𝑡)
1817ssdifssd 4105 . . . . . . . . . . 11 (𝑧𝑃 → ((𝑡𝑧) ∖ ran 𝑈) ⊆ ran 𝑡)
19 sseq1 3978 . . . . . . . . . . 11 (𝑎 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎 ran 𝑡 ↔ ((𝑡𝑧) ∖ ran 𝑈) ⊆ ran 𝑡))
2018, 19syl5ibrcom 250 . . . . . . . . . 10 (𝑧𝑃 → (𝑎 = ((𝑡𝑧) ∖ ran 𝑈) → 𝑎 ran 𝑡))
2120rexlimiv 3272 . . . . . . . . 9 (∃𝑧𝑃 𝑎 = ((𝑡𝑧) ∖ ran 𝑈) → 𝑎 ran 𝑡)
2215, 21sylbi 220 . . . . . . . 8 (𝑎 ∈ {𝑎 ∣ ∃𝑧𝑃 𝑎 = ((𝑡𝑧) ∖ ran 𝑈)} → 𝑎 ran 𝑡)
23 eqid 2824 . . . . . . . . 9 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
2423rnmpt 5814 . . . . . . . 8 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑎 ∣ ∃𝑧𝑃 𝑎 = ((𝑡𝑧) ∖ ran 𝑈)}
2522, 24eleq2s 2934 . . . . . . 7 (𝑎 ∈ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) → 𝑎 ran 𝑡)
2614, 25mprgbir 3148 . . . . . 6 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ⊆ ran 𝑡
2713, 26sstri 3962 . . . . 5 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran 𝑡
2811, 27eqsstrdi 4007 . . . 4 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 ran 𝑡)
2928adantl 485 . . 3 ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) → ran 𝑍 ran 𝑡)
309, 29jaoi 854 . 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 399  wo 844   = wceq 1538  wcel 2115  {cab 2802  wral 3133  wrex 3134  {crab 3137  Vcvv 3480  cdif 3916  cin 3918  wss 3919  c0 4276  ifcif 4450  𝒫 cpw 4522   cuni 4824   cint 4862   class class class wbr 5052  cmpt 5132  ran crn 5543  ccom 5546  suc csuc 6180  cfv 6343  crio 7106  (class class class)co 7149  cmpo 7151  ωcom 7574  seqωcseqom 8079  m cmap 8402  cen 8502  Fincfn 8505
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fv 6351
This theorem is referenced by:  fin23lem31  9763
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