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Theorem fin23lem16 10375
Description: Lemma for fin23 10429. 𝑈 ranges over the original set; in particular ran 𝑈 is a set, although we do not assume here that 𝑈 is. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Hypothesis
Ref Expression
fin23lem.a 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
Assertion
Ref Expression
fin23lem16 ran 𝑈 = ran 𝑡
Distinct variable groups:   𝑡,𝑖,𝑢   𝑈,𝑖,𝑢
Allowed substitution hint:   𝑈(𝑡)

Proof of Theorem fin23lem16
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unissb 4939 . . 3 ( ran 𝑈 ran 𝑡 ↔ ∀𝑎 ∈ ran 𝑈 𝑎 ran 𝑡)
2 fin23lem.a . . . . . 6 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
32fnseqom 8495 . . . . 5 𝑈 Fn ω
4 fvelrnb 6969 . . . . 5 (𝑈 Fn ω → (𝑎 ∈ ran 𝑈 ↔ ∃𝑏 ∈ ω (𝑈𝑏) = 𝑎))
53, 4ax-mp 5 . . . 4 (𝑎 ∈ ran 𝑈 ↔ ∃𝑏 ∈ ω (𝑈𝑏) = 𝑎)
6 peano1 7910 . . . . . . . 8 ∅ ∈ ω
7 0ss 4400 . . . . . . . . 9 ∅ ⊆ 𝑏
82fin23lem15 10374 . . . . . . . . 9 (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈𝑏) ⊆ (𝑈‘∅))
97, 8mpan2 691 . . . . . . . 8 ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈𝑏) ⊆ (𝑈‘∅))
106, 9mpan2 691 . . . . . . 7 (𝑏 ∈ ω → (𝑈𝑏) ⊆ (𝑈‘∅))
11 vex 3484 . . . . . . . . . 10 𝑡 ∈ V
1211rnex 7932 . . . . . . . . 9 ran 𝑡 ∈ V
1312uniex 7761 . . . . . . . 8 ran 𝑡 ∈ V
142seqom0g 8496 . . . . . . . 8 ( ran 𝑡 ∈ V → (𝑈‘∅) = ran 𝑡)
1513, 14ax-mp 5 . . . . . . 7 (𝑈‘∅) = ran 𝑡
1610, 15sseqtrdi 4024 . . . . . 6 (𝑏 ∈ ω → (𝑈𝑏) ⊆ ran 𝑡)
17 sseq1 4009 . . . . . 6 ((𝑈𝑏) = 𝑎 → ((𝑈𝑏) ⊆ ran 𝑡𝑎 ran 𝑡))
1816, 17syl5ibcom 245 . . . . 5 (𝑏 ∈ ω → ((𝑈𝑏) = 𝑎𝑎 ran 𝑡))
1918rexlimiv 3148 . . . 4 (∃𝑏 ∈ ω (𝑈𝑏) = 𝑎𝑎 ran 𝑡)
205, 19sylbi 217 . . 3 (𝑎 ∈ ran 𝑈𝑎 ran 𝑡)
211, 20mprgbir 3068 . 2 ran 𝑈 ran 𝑡
22 fnfvelrn 7100 . . . . 5 ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈)
233, 6, 22mp2an 692 . . . 4 (𝑈‘∅) ∈ ran 𝑈
2415, 23eqeltrri 2838 . . 3 ran 𝑡 ∈ ran 𝑈
25 elssuni 4937 . . 3 ( ran 𝑡 ∈ ran 𝑈 ran 𝑡 ran 𝑈)
2624, 25ax-mp 5 . 2 ran 𝑡 ran 𝑈
2721, 26eqssi 4000 1 ran 𝑈 = ran 𝑡
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cin 3950  wss 3951  c0 4333  ifcif 4525   cuni 4907  ran crn 5686   Fn wfn 6556  cfv 6561  cmpo 7433  ωcom 7887  seqωcseqom 8487
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  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488
This theorem is referenced by:  fin23lem17  10378  fin23lem31  10383
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