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Theorem trclimalb2 38538
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 3352 . . . 4 (𝑅𝑉𝑅 ∈ V)
21adantr 472 . . 3 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → 𝑅 ∈ V)
3 oveq1 6821 . . . . . . 7 (𝑟 = 𝑅 → (𝑟𝑟𝑘) = (𝑅𝑟𝑘))
43iuneq2d 4699 . . . . . 6 (𝑟 = 𝑅 𝑘 ∈ ℕ (𝑟𝑟𝑘) = 𝑘 ∈ ℕ (𝑅𝑟𝑘))
5 dftrcl3 38532 . . . . . 6 t+ = (𝑟 ∈ V ↦ 𝑘 ∈ ℕ (𝑟𝑟𝑘))
6 nnex 11238 . . . . . . 7 ℕ ∈ V
7 ovex 6842 . . . . . . 7 (𝑅𝑟𝑘) ∈ V
86, 7iunex 7313 . . . . . 6 𝑘 ∈ ℕ (𝑅𝑟𝑘) ∈ V
94, 5, 8fvmpt 6445 . . . . 5 (𝑅 ∈ V → (t+‘𝑅) = 𝑘 ∈ ℕ (𝑅𝑟𝑘))
109imaeq1d 5623 . . . 4 (𝑅 ∈ V → ((t+‘𝑅) “ 𝐴) = ( 𝑘 ∈ ℕ (𝑅𝑟𝑘) “ 𝐴))
11 imaiun1 38463 . . . 4 ( 𝑘 ∈ ℕ (𝑅𝑟𝑘) “ 𝐴) = 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴)
1210, 11syl6eq 2810 . . 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 6822 . . . . . . . . 9 (𝑥 = 1 → (𝑅𝑟𝑥) = (𝑅𝑟1))
1514imaeq1d 5623 . . . . . . . 8 (𝑥 = 1 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟1) “ 𝐴))
1615sseq1d 3773 . . . . . . 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 6822 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑅𝑟𝑥) = (𝑅𝑟𝑦))
1918imaeq1d 5623 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟𝑦) “ 𝐴))
2019sseq1d 3773 . . . . . . 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 6822 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑅𝑟𝑥) = (𝑅𝑟(𝑦 + 1)))
2322imaeq1d 5623 . . . . . . . 8 (𝑥 = (𝑦 + 1) → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟(𝑦 + 1)) “ 𝐴))
2423sseq1d 3773 . . . . . . 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 6822 . . . . . . . . 9 (𝑥 = 𝑘 → (𝑅𝑟𝑥) = (𝑅𝑟𝑘))
2726imaeq1d 5623 . . . . . . . 8 (𝑥 = 𝑘 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟𝑘) “ 𝐴))
2827sseq1d 3773 . . . . . . 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 13985 . . . . . . . . 9 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
3130imaeq1d 5623 . . . . . . . 8 (𝑅𝑉 → ((𝑅𝑟1) “ 𝐴) = (𝑅𝐴))
3231adantr 472 . . . . . . 7 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟1) “ 𝐴) = (𝑅𝐴))
33 ssun1 3919 . . . . . . . . 9 𝐴 ⊆ (𝐴𝐵)
34 imass2 5659 . . . . . . . . 9 (𝐴 ⊆ (𝐴𝐵) → (𝑅𝐴) ⊆ (𝑅 “ (𝐴𝐵)))
3533, 34mp1i 13 . . . . . . . 8 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑅𝐴) ⊆ (𝑅 “ (𝐴𝐵)))
36 simpr 479 . . . . . . . 8 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑅 “ (𝐴𝐵)) ⊆ 𝐵)
3735, 36sstrd 3754 . . . . . . 7 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑅𝐴) ⊆ 𝐵)
3832, 37eqsstrd 3780 . . . . . 6 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟1) “ 𝐴) ⊆ 𝐵)
39 simp2l 1242 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑅𝑉)
40 simp1 1131 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑦 ∈ ℕ)
41 relexpsucnnl 13991 . . . . . . . . . . . 12 ((𝑅𝑉𝑦 ∈ ℕ) → (𝑅𝑟(𝑦 + 1)) = (𝑅 ∘ (𝑅𝑟𝑦)))
4241imaeq1d 5623 . . . . . . . . . . 11 ((𝑅𝑉𝑦 ∈ ℕ) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = ((𝑅 ∘ (𝑅𝑟𝑦)) “ 𝐴))
43 imaco 5801 . . . . . . . . . . 11 ((𝑅 ∘ (𝑅𝑟𝑦)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴))
4442, 43syl6eq 2810 . . . . . . . . . 10 ((𝑅𝑉𝑦 ∈ ℕ) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)))
4539, 40, 44syl2anc 696 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)))
46 imass2 5659 . . . . . . . . . . 11 (((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵 → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ (𝑅𝐵))
47463ad2ant3 1130 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ (𝑅𝐵))
48 ssun2 3920 . . . . . . . . . . . 12 𝐵 ⊆ (𝐴𝐵)
49 imass2 5659 . . . . . . . . . . . 12 (𝐵 ⊆ (𝐴𝐵) → (𝑅𝐵) ⊆ (𝑅 “ (𝐴𝐵)))
5048, 49mp1i 13 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅𝐵) ⊆ (𝑅 “ (𝐴𝐵)))
51 simp2r 1243 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ (𝐴𝐵)) ⊆ 𝐵)
5250, 51sstrd 3754 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅𝐵) ⊆ 𝐵)
5347, 52sstrd 3754 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ 𝐵)
5445, 53eqsstrd 3780 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)
55543exp 1113 . . . . . . 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 11250 . . . . 5 (𝑘 ∈ ℕ → ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵))
5857com12 32 . . . 4 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑘 ∈ ℕ → ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵))
5958ralrimiv 3103 . . 3 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ∀𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)
60 iunss 4713 . . 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 3780 1 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cun 3713  wss 3715   ciun 4672  cima 5269  ccom 5270  cfv 6049  (class class class)co 6814  1c1 10149   + caddc 10151  cn 11232  t+ctcl 13945  𝑟crelexp 13979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-n0 11505  df-z 11590  df-uz 11900  df-seq 13016  df-trcl 13947  df-relexp 13980
This theorem is referenced by:  brtrclfv2  38539  frege77d  38558
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