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Theorem fin23lem29 10378
Description: Lemma for fin23 10426. 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 4571 . . 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 5949 . . . . . 6 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
54unieqd 4924 . . . . 5 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6 rncoss 5988 . . . . . 6 ran (𝑡𝑅) ⊆ ran 𝑡
76unissi 4920 . . . . 5 ran (𝑡𝑅) ⊆ ran 𝑡
85, 7eqsstrdi 4049 . . . 4 (𝑍 = (𝑡𝑅) → ran 𝑍 ran 𝑡)
98adantl 481 . . 3 ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) → ran 𝑍 ran 𝑡)
10 rneq 5949 . . . . . 6 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
1110unieqd 4924 . . . . 5 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
12 rncoss 5988 . . . . . . 7 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
1312unissi 4920 . . . . . 6 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
14 unissb 4943 . . . . . . 7 ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ⊆ ran 𝑡 ↔ ∀𝑎 ∈ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))𝑎 ran 𝑡)
15 abid 2715 . . . . . . . . 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 4156 . . . . . . . . . . 11 (𝑧𝑃 → ((𝑡𝑧) ∖ ran 𝑈) ⊆ ran 𝑡)
19 sseq1 4020 . . . . . . . . . . 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 3145 . . . . . . . . 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 2734 . . . . . . . . 9 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
2423rnmpt 5970 . . . . . . . 8 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑎 ∣ ∃𝑧𝑃 𝑎 = ((𝑡𝑧) ∖ ran 𝑈)}
2522, 24eleq2s 2856 . . . . . . 7 (𝑎 ∈ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) → 𝑎 ran 𝑡)
2614, 25mprgbir 3065 . . . . . 6 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ⊆ ran 𝑡
2713, 26sstri 4004 . . . . 5 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran 𝑡
2811, 27eqsstrdi 4049 . . . 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 857 . 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 847   = wceq 1536  wcel 2105  {cab 2711  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  cdif 3959  cin 3961  wss 3962  c0 4338  ifcif 4530  𝒫 cpw 4604   cuni 4911   cint 4950   class class class wbr 5147  cmpt 5230  ran crn 5689  ccom 5692  suc csuc 6387  cfv 6562  crio 7386  (class class class)co 7430  cmpo 7432  ωcom 7886  seqωcseqom 8485  m cmap 8864  cen 8980  Fincfn 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fv 6570
This theorem is referenced by:  fin23lem31  10380
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