Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trclimalb2 Structured version   Visualization version   GIF version

Theorem trclimalb2 38335
Description: Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.)
Assertion
Ref Expression
trclimalb2 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)

Proof of Theorem trclimalb2
Dummy variables 𝑥 𝑘 𝑦 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3243 . . . 4 (𝑅𝑉𝑅 ∈ V)
21adantr 480 . . 3 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → 𝑅 ∈ V)
3 oveq1 6697 . . . . . . 7 (𝑟 = 𝑅 → (𝑟𝑟𝑘) = (𝑅𝑟𝑘))
43iuneq2d 4579 . . . . . 6 (𝑟 = 𝑅 𝑘 ∈ ℕ (𝑟𝑟𝑘) = 𝑘 ∈ ℕ (𝑅𝑟𝑘))
5 dftrcl3 38329 . . . . . 6 t+ = (𝑟 ∈ V ↦ 𝑘 ∈ ℕ (𝑟𝑟𝑘))
6 nnex 11064 . . . . . . 7 ℕ ∈ V
7 ovex 6718 . . . . . . 7 (𝑅𝑟𝑘) ∈ V
86, 7iunex 7189 . . . . . 6 𝑘 ∈ ℕ (𝑅𝑟𝑘) ∈ V
94, 5, 8fvmpt 6321 . . . . 5 (𝑅 ∈ V → (t+‘𝑅) = 𝑘 ∈ ℕ (𝑅𝑟𝑘))
109imaeq1d 5500 . . . 4 (𝑅 ∈ V → ((t+‘𝑅) “ 𝐴) = ( 𝑘 ∈ ℕ (𝑅𝑟𝑘) “ 𝐴))
11 imaiun1 38260 . . . 4 ( 𝑘 ∈ ℕ (𝑅𝑟𝑘) “ 𝐴) = 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴)
1210, 11syl6eq 2701 . . 3 (𝑅 ∈ V → ((t+‘𝑅) “ 𝐴) = 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴))
132, 12syl 17 . 2 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) = 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴))
14 oveq2 6698 . . . . . . . . 9 (𝑥 = 1 → (𝑅𝑟𝑥) = (𝑅𝑟1))
1514imaeq1d 5500 . . . . . . . 8 (𝑥 = 1 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟1) “ 𝐴))
1615sseq1d 3665 . . . . . . 7 (𝑥 = 1 → (((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅𝑟1) “ 𝐴) ⊆ 𝐵))
1716imbi2d 329 . . . . . 6 (𝑥 = 1 → (((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟1) “ 𝐴) ⊆ 𝐵)))
18 oveq2 6698 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑅𝑟𝑥) = (𝑅𝑟𝑦))
1918imaeq1d 5500 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟𝑦) “ 𝐴))
2019sseq1d 3665 . . . . . . 7 (𝑥 = 𝑦 → (((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵))
2120imbi2d 329 . . . . . 6 (𝑥 = 𝑦 → (((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵)))
22 oveq2 6698 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑅𝑟𝑥) = (𝑅𝑟(𝑦 + 1)))
2322imaeq1d 5500 . . . . . . . 8 (𝑥 = (𝑦 + 1) → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟(𝑦 + 1)) “ 𝐴))
2423sseq1d 3665 . . . . . . 7 (𝑥 = (𝑦 + 1) → (((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵))
2524imbi2d 329 . . . . . 6 (𝑥 = (𝑦 + 1) → (((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)))
26 oveq2 6698 . . . . . . . . 9 (𝑥 = 𝑘 → (𝑅𝑟𝑥) = (𝑅𝑟𝑘))
2726imaeq1d 5500 . . . . . . . 8 (𝑥 = 𝑘 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟𝑘) “ 𝐴))
2827sseq1d 3665 . . . . . . 7 (𝑥 = 𝑘 → (((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵))
2928imbi2d 329 . . . . . 6 (𝑥 = 𝑘 → (((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)))
30 relexp1g 13810 . . . . . . . . 9 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
3130imaeq1d 5500 . . . . . . . 8 (𝑅𝑉 → ((𝑅𝑟1) “ 𝐴) = (𝑅𝐴))
3231adantr 480 . . . . . . 7 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟1) “ 𝐴) = (𝑅𝐴))
33 ssun1 3809 . . . . . . . . 9 𝐴 ⊆ (𝐴𝐵)
34 imass2 5536 . . . . . . . . 9 (𝐴 ⊆ (𝐴𝐵) → (𝑅𝐴) ⊆ (𝑅 “ (𝐴𝐵)))
3533, 34mp1i 13 . . . . . . . 8 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑅𝐴) ⊆ (𝑅 “ (𝐴𝐵)))
36 simpr 476 . . . . . . . 8 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑅 “ (𝐴𝐵)) ⊆ 𝐵)
3735, 36sstrd 3646 . . . . . . 7 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑅𝐴) ⊆ 𝐵)
3832, 37eqsstrd 3672 . . . . . 6 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟1) “ 𝐴) ⊆ 𝐵)
39 simp2l 1107 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑅𝑉)
40 simp1 1081 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑦 ∈ ℕ)
41 relexpsucnnl 13816 . . . . . . . . . . . 12 ((𝑅𝑉𝑦 ∈ ℕ) → (𝑅𝑟(𝑦 + 1)) = (𝑅 ∘ (𝑅𝑟𝑦)))
4241imaeq1d 5500 . . . . . . . . . . 11 ((𝑅𝑉𝑦 ∈ ℕ) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = ((𝑅 ∘ (𝑅𝑟𝑦)) “ 𝐴))
43 imaco 5678 . . . . . . . . . . 11 ((𝑅 ∘ (𝑅𝑟𝑦)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴))
4442, 43syl6eq 2701 . . . . . . . . . 10 ((𝑅𝑉𝑦 ∈ ℕ) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)))
4539, 40, 44syl2anc 694 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)))
46 imass2 5536 . . . . . . . . . . 11 (((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵 → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ (𝑅𝐵))
47463ad2ant3 1104 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ (𝑅𝐵))
48 ssun2 3810 . . . . . . . . . . . 12 𝐵 ⊆ (𝐴𝐵)
49 imass2 5536 . . . . . . . . . . . 12 (𝐵 ⊆ (𝐴𝐵) → (𝑅𝐵) ⊆ (𝑅 “ (𝐴𝐵)))
5048, 49mp1i 13 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅𝐵) ⊆ (𝑅 “ (𝐴𝐵)))
51 simp2r 1108 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ (𝐴𝐵)) ⊆ 𝐵)
5250, 51sstrd 3646 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅𝐵) ⊆ 𝐵)
5347, 52sstrd 3646 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ 𝐵)
5445, 53eqsstrd 3672 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)
55543exp 1283 . . . . . . 7 (𝑦 ∈ ℕ → ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵 → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)))
5655a2d 29 . . . . . 6 (𝑦 ∈ ℕ → (((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)))
5717, 21, 25, 29, 38, 56nnind 11076 . . . . 5 (𝑘 ∈ ℕ → ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵))
5857com12 32 . . . 4 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑘 ∈ ℕ → ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵))
5958ralrimiv 2994 . . 3 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ∀𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)
60 iunss 4593 . . 3 ( 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵 ↔ ∀𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)
6159, 60sylibr 224 . 2 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)
6213, 61eqsstrd 3672 1 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cun 3605  wss 3607   ciun 4552  cima 5146  ccom 5147  cfv 5926  (class class class)co 6690  1c1 9975   + caddc 9977  cn 11058  t+ctcl 13770  𝑟crelexp 13804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-seq 12842  df-trcl 13772  df-relexp 13805
This theorem is referenced by:  brtrclfv2  38336  frege77d  38355
  Copyright terms: Public domain W3C validator