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Theorem fin23lem16 9358
Description: Lemma for fin23 9412. 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 4603 . . 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 7702 . . . . 5 𝑈 Fn ω
4 fvelrnb 6385 . . . . 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 7231 . . . . . . . 8 ∅ ∈ ω
7 0ss 4114 . . . . . . . . 9 ∅ ⊆ 𝑏
82fin23lem15 9357 . . . . . . . . 9 (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈𝑏) ⊆ (𝑈‘∅))
97, 8mpan2 663 . . . . . . . 8 ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈𝑏) ⊆ (𝑈‘∅))
106, 9mpan2 663 . . . . . . 7 (𝑏 ∈ ω → (𝑈𝑏) ⊆ (𝑈‘∅))
11 vex 3352 . . . . . . . . . 10 𝑡 ∈ V
1211rnex 7246 . . . . . . . . 9 ran 𝑡 ∈ V
1312uniex 7099 . . . . . . . 8 ran 𝑡 ∈ V
142seqom0g 7703 . . . . . . . 8 ( ran 𝑡 ∈ V → (𝑈‘∅) = ran 𝑡)
1513, 14ax-mp 5 . . . . . . 7 (𝑈‘∅) = ran 𝑡
1610, 15syl6sseq 3798 . . . . . 6 (𝑏 ∈ ω → (𝑈𝑏) ⊆ ran 𝑡)
17 sseq1 3773 . . . . . 6 ((𝑈𝑏) = 𝑎 → ((𝑈𝑏) ⊆ ran 𝑡𝑎 ran 𝑡))
1816, 17syl5ibcom 235 . . . . 5 (𝑏 ∈ ω → ((𝑈𝑏) = 𝑎𝑎 ran 𝑡))
1918rexlimiv 3174 . . . 4 (∃𝑏 ∈ ω (𝑈𝑏) = 𝑎𝑎 ran 𝑡)
205, 19sylbi 207 . . 3 (𝑎 ∈ ran 𝑈𝑎 ran 𝑡)
211, 20mprgbir 3075 . 2 ran 𝑈 ran 𝑡
22 fnfvelrn 6499 . . . . 5 ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈)
233, 6, 22mp2an 664 . . . 4 (𝑈‘∅) ∈ ran 𝑈
2415, 23eqeltrri 2846 . . 3 ran 𝑡 ∈ ran 𝑈
25 elssuni 4601 . . 3 ( ran 𝑡 ∈ ran 𝑈 ran 𝑡 ran 𝑈)
2624, 25ax-mp 5 . 2 ran 𝑡 ran 𝑈
2721, 26eqssi 3766 1 ran 𝑈 = ran 𝑡
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1630  wcel 2144  wrex 3061  Vcvv 3349  cin 3720  wss 3721  c0 4061  ifcif 4223   cuni 4572  ran crn 5250   Fn wfn 6026  cfv 6031  cmpt2 6794  ωcom 7211  seq𝜔cseqom 7694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-seqom 7695
This theorem is referenced by:  fin23lem17  9361  fin23lem31  9366
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