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Theorem fin23lem16 9104
 Description: Lemma for fin23 9158. 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 4437 . . 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 7498 . . . . 5 𝑈 Fn ω
4 fvelrnb 6202 . . . . 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 7035 . . . . . . . 8 ∅ ∈ ω
7 0ss 3946 . . . . . . . . 9 ∅ ⊆ 𝑏
82fin23lem15 9103 . . . . . . . . 9 (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈𝑏) ⊆ (𝑈‘∅))
97, 8mpan2 706 . . . . . . . 8 ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈𝑏) ⊆ (𝑈‘∅))
106, 9mpan2 706 . . . . . . 7 (𝑏 ∈ ω → (𝑈𝑏) ⊆ (𝑈‘∅))
11 vex 3189 . . . . . . . . . 10 𝑡 ∈ V
1211rnex 7050 . . . . . . . . 9 ran 𝑡 ∈ V
1312uniex 6909 . . . . . . . 8 ran 𝑡 ∈ V
142seqom0g 7499 . . . . . . . 8 ( ran 𝑡 ∈ V → (𝑈‘∅) = ran 𝑡)
1513, 14ax-mp 5 . . . . . . 7 (𝑈‘∅) = ran 𝑡
1610, 15syl6sseq 3632 . . . . . 6 (𝑏 ∈ ω → (𝑈𝑏) ⊆ ran 𝑡)
17 sseq1 3607 . . . . . 6 ((𝑈𝑏) = 𝑎 → ((𝑈𝑏) ⊆ ran 𝑡𝑎 ran 𝑡))
1816, 17syl5ibcom 235 . . . . 5 (𝑏 ∈ ω → ((𝑈𝑏) = 𝑎𝑎 ran 𝑡))
1918rexlimiv 3020 . . . 4 (∃𝑏 ∈ ω (𝑈𝑏) = 𝑎𝑎 ran 𝑡)
205, 19sylbi 207 . . 3 (𝑎 ∈ ran 𝑈𝑎 ran 𝑡)
211, 20mprgbir 2922 . 2 ran 𝑈 ran 𝑡
22 fnfvelrn 6314 . . . . 5 ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈)
233, 6, 22mp2an 707 . . . 4 (𝑈‘∅) ∈ ran 𝑈
2415, 23eqeltrri 2695 . . 3 ran 𝑡 ∈ ran 𝑈
25 elssuni 4435 . . 3 ( ran 𝑡 ∈ ran 𝑈 ran 𝑡 ran 𝑈)
2624, 25ax-mp 5 . 2 ran 𝑡 ran 𝑈
2721, 26eqssi 3600 1 ran 𝑈 = ran 𝑡
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2908  Vcvv 3186   ∩ cin 3555   ⊆ wss 3556  ∅c0 3893  ifcif 4060  ∪ cuni 4404  ran crn 5077   Fn wfn 5844  ‘cfv 5849   ↦ cmpt2 6609  ωcom 7015  seq𝜔cseqom 7490 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-seqom 7491 This theorem is referenced by:  fin23lem17  9107  fin23lem31  9112
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