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Theorem trclimalb2 37037
 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 3185 . . . 4 (𝑅𝑉𝑅 ∈ V)
21adantr 480 . . 3 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → 𝑅 ∈ V)
3 oveq1 6556 . . . . . . 7 (𝑟 = 𝑅 → (𝑟𝑟𝑘) = (𝑅𝑟𝑘))
43iuneq2d 4483 . . . . . 6 (𝑟 = 𝑅 𝑘 ∈ ℕ (𝑟𝑟𝑘) = 𝑘 ∈ ℕ (𝑅𝑟𝑘))
5 dftrcl3 37031 . . . . . 6 t+ = (𝑟 ∈ V ↦ 𝑘 ∈ ℕ (𝑟𝑟𝑘))
6 nnex 10903 . . . . . . 7 ℕ ∈ V
7 ovex 6577 . . . . . . 7 (𝑅𝑟𝑘) ∈ V
86, 7iunex 7039 . . . . . 6 𝑘 ∈ ℕ (𝑅𝑟𝑘) ∈ V
94, 5, 8fvmpt 6191 . . . . 5 (𝑅 ∈ V → (t+‘𝑅) = 𝑘 ∈ ℕ (𝑅𝑟𝑘))
109imaeq1d 5384 . . . 4 (𝑅 ∈ V → ((t+‘𝑅) “ 𝐴) = ( 𝑘 ∈ ℕ (𝑅𝑟𝑘) “ 𝐴))
11 imaiun1 36962 . . . 4 ( 𝑘 ∈ ℕ (𝑅𝑟𝑘) “ 𝐴) = 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴)
1210, 11syl6eq 2660 . . 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 6557 . . . . . . . . 9 (𝑥 = 1 → (𝑅𝑟𝑥) = (𝑅𝑟1))
1514imaeq1d 5384 . . . . . . . 8 (𝑥 = 1 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟1) “ 𝐴))
1615sseq1d 3595 . . . . . . 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 6557 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑅𝑟𝑥) = (𝑅𝑟𝑦))
1918imaeq1d 5384 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟𝑦) “ 𝐴))
2019sseq1d 3595 . . . . . . 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 6557 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑅𝑟𝑥) = (𝑅𝑟(𝑦 + 1)))
2322imaeq1d 5384 . . . . . . . 8 (𝑥 = (𝑦 + 1) → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟(𝑦 + 1)) “ 𝐴))
2423sseq1d 3595 . . . . . . 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 6557 . . . . . . . . 9 (𝑥 = 𝑘 → (𝑅𝑟𝑥) = (𝑅𝑟𝑘))
2726imaeq1d 5384 . . . . . . . 8 (𝑥 = 𝑘 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟𝑘) “ 𝐴))
2827sseq1d 3595 . . . . . . 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 13614 . . . . . . . . 9 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
3130imaeq1d 5384 . . . . . . . 8 (𝑅𝑉 → ((𝑅𝑟1) “ 𝐴) = (𝑅𝐴))
3231adantr 480 . . . . . . 7 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟1) “ 𝐴) = (𝑅𝐴))
33 ssun1 3738 . . . . . . . . 9 𝐴 ⊆ (𝐴𝐵)
34 imass2 5420 . . . . . . . . 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 3578 . . . . . . 7 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑅𝐴) ⊆ 𝐵)
3832, 37eqsstrd 3602 . . . . . 6 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟1) “ 𝐴) ⊆ 𝐵)
39 simp2l 1080 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑅𝑉)
40 simp1 1054 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑦 ∈ ℕ)
41 relexpsucnnl 13620 . . . . . . . . . . . 12 ((𝑅𝑉𝑦 ∈ ℕ) → (𝑅𝑟(𝑦 + 1)) = (𝑅 ∘ (𝑅𝑟𝑦)))
4241imaeq1d 5384 . . . . . . . . . . 11 ((𝑅𝑉𝑦 ∈ ℕ) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = ((𝑅 ∘ (𝑅𝑟𝑦)) “ 𝐴))
43 imaco 5557 . . . . . . . . . . 11 ((𝑅 ∘ (𝑅𝑟𝑦)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴))
4442, 43syl6eq 2660 . . . . . . . . . 10 ((𝑅𝑉𝑦 ∈ ℕ) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)))
4539, 40, 44syl2anc 691 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)))
46 imass2 5420 . . . . . . . . . . 11 (((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵 → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ (𝑅𝐵))
47463ad2ant3 1077 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ (𝑅𝐵))
48 ssun2 3739 . . . . . . . . . . . 12 𝐵 ⊆ (𝐴𝐵)
49 imass2 5420 . . . . . . . . . . . 12 (𝐵 ⊆ (𝐴𝐵) → (𝑅𝐵) ⊆ (𝑅 “ (𝐴𝐵)))
5048, 49mp1i 13 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅𝐵) ⊆ (𝑅 “ (𝐴𝐵)))
51 simp2r 1081 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ (𝐴𝐵)) ⊆ 𝐵)
5250, 51sstrd 3578 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅𝐵) ⊆ 𝐵)
5347, 52sstrd 3578 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ 𝐵)
5445, 53eqsstrd 3602 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)
55543exp 1256 . . . . . . 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 10915 . . . . 5 (𝑘 ∈ ℕ → ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵))
5857com12 32 . . . 4 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑘 ∈ ℕ → ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵))
5958ralrimiv 2948 . . 3 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ∀𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)
60 iunss 4497 . . 3 ( 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵 ↔ ∀𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)
6159, 60sylibr 223 . 2 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)
6213, 61eqsstrd 3602 1 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∪ ciun 4455   “ cima 5041   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818  ℕcn 10897  t+ctcl 13572  ↑𝑟crelexp 13608 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-trcl 13574  df-relexp 13609 This theorem is referenced by:  brtrclfv2  37038  frege77d  37057
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