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Theorem fin23lem14 9107
Description: Lemma for fin23 9163. 𝑈 will never evolve to an empty set if it did not start with one. (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
fin23lem14 ((𝐴 ∈ ω ∧ ran 𝑡 ≠ ∅) → (𝑈𝐴) ≠ ∅)
Distinct variable groups:   𝑡,𝑖,𝑢   𝐴,𝑖,𝑢   𝑈,𝑖,𝑢
Allowed substitution hints:   𝐴(𝑡)   𝑈(𝑡)

Proof of Theorem fin23lem14
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6153 . . . . 5 (𝑎 = ∅ → (𝑈𝑎) = (𝑈‘∅))
21neeq1d 2849 . . . 4 (𝑎 = ∅ → ((𝑈𝑎) ≠ ∅ ↔ (𝑈‘∅) ≠ ∅))
32imbi2d 330 . . 3 (𝑎 = ∅ → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈‘∅) ≠ ∅)))
4 fveq2 6153 . . . . 5 (𝑎 = 𝑏 → (𝑈𝑎) = (𝑈𝑏))
54neeq1d 2849 . . . 4 (𝑎 = 𝑏 → ((𝑈𝑎) ≠ ∅ ↔ (𝑈𝑏) ≠ ∅))
65imbi2d 330 . . 3 (𝑎 = 𝑏 → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈𝑏) ≠ ∅)))
7 fveq2 6153 . . . . 5 (𝑎 = suc 𝑏 → (𝑈𝑎) = (𝑈‘suc 𝑏))
87neeq1d 2849 . . . 4 (𝑎 = suc 𝑏 → ((𝑈𝑎) ≠ ∅ ↔ (𝑈‘suc 𝑏) ≠ ∅))
98imbi2d 330 . . 3 (𝑎 = suc 𝑏 → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈‘suc 𝑏) ≠ ∅)))
10 fveq2 6153 . . . . 5 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
1110neeq1d 2849 . . . 4 (𝑎 = 𝐴 → ((𝑈𝑎) ≠ ∅ ↔ (𝑈𝐴) ≠ ∅))
1211imbi2d 330 . . 3 (𝑎 = 𝐴 → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈𝐴) ≠ ∅)))
13 vex 3192 . . . . . . 7 𝑡 ∈ V
1413rnex 7054 . . . . . 6 ran 𝑡 ∈ V
1514uniex 6913 . . . . 5 ran 𝑡 ∈ V
16 fin23lem.a . . . . . 6 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
1716seqom0g 7503 . . . . 5 ( ran 𝑡 ∈ V → (𝑈‘∅) = ran 𝑡)
1815, 17mp1i 13 . . . 4 ( ran 𝑡 ≠ ∅ → (𝑈‘∅) = ran 𝑡)
19 id 22 . . . 4 ( ran 𝑡 ≠ ∅ → ran 𝑡 ≠ ∅)
2018, 19eqnetrd 2857 . . 3 ( ran 𝑡 ≠ ∅ → (𝑈‘∅) ≠ ∅)
2116fin23lem12 9105 . . . . . . 7 (𝑏 ∈ ω → (𝑈‘suc 𝑏) = if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))))
2221adantr 481 . . . . . 6 ((𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅) → (𝑈‘suc 𝑏) = if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))))
23 iftrue 4069 . . . . . . . . 9 (((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = (𝑈𝑏))
2423adantr 481 . . . . . . . 8 ((((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = (𝑈𝑏))
25 simprr 795 . . . . . . . 8 ((((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → (𝑈𝑏) ≠ ∅)
2624, 25eqnetrd 2857 . . . . . . 7 ((((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) ≠ ∅)
27 iffalse 4072 . . . . . . . . 9 (¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = ((𝑡𝑏) ∩ (𝑈𝑏)))
2827adantr 481 . . . . . . . 8 ((¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = ((𝑡𝑏) ∩ (𝑈𝑏)))
29 df-ne 2791 . . . . . . . . . 10 (((𝑡𝑏) ∩ (𝑈𝑏)) ≠ ∅ ↔ ¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅)
3029biimpri 218 . . . . . . . . 9 (¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ → ((𝑡𝑏) ∩ (𝑈𝑏)) ≠ ∅)
3130adantr 481 . . . . . . . 8 ((¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → ((𝑡𝑏) ∩ (𝑈𝑏)) ≠ ∅)
3228, 31eqnetrd 2857 . . . . . . 7 ((¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) ≠ ∅)
3326, 32pm2.61ian 830 . . . . . 6 ((𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) ≠ ∅)
3422, 33eqnetrd 2857 . . . . 5 ((𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅) → (𝑈‘suc 𝑏) ≠ ∅)
3534ex 450 . . . 4 (𝑏 ∈ ω → ((𝑈𝑏) ≠ ∅ → (𝑈‘suc 𝑏) ≠ ∅))
3635imim2d 57 . . 3 (𝑏 ∈ ω → (( ran 𝑡 ≠ ∅ → (𝑈𝑏) ≠ ∅) → ( ran 𝑡 ≠ ∅ → (𝑈‘suc 𝑏) ≠ ∅)))
373, 6, 9, 12, 20, 36finds 7046 . 2 (𝐴 ∈ ω → ( ran 𝑡 ≠ ∅ → (𝑈𝐴) ≠ ∅))
3837imp 445 1 ((𝐴 ∈ ω ∧ ran 𝑡 ≠ ∅) → (𝑈𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3189  cin 3558  c0 3896  ifcif 4063   cuni 4407  ran crn 5080  suc csuc 5689  cfv 5852  cmpt2 6612  ωcom 7019  seq𝜔cseqom 7494
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 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-seqom 7495
This theorem is referenced by:  fin23lem21  9113
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