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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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