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Theorem trclimalb2 40414
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 3462 . . . 4 (𝑅𝑉𝑅 ∈ V)
21adantr 484 . . 3 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → 𝑅 ∈ V)
3 oveq1 7146 . . . . . . 7 (𝑟 = 𝑅 → (𝑟𝑟𝑘) = (𝑅𝑟𝑘))
43iuneq2d 4913 . . . . . 6 (𝑟 = 𝑅 𝑘 ∈ ℕ (𝑟𝑟𝑘) = 𝑘 ∈ ℕ (𝑅𝑟𝑘))
5 dftrcl3 40408 . . . . . 6 t+ = (𝑟 ∈ V ↦ 𝑘 ∈ ℕ (𝑟𝑟𝑘))
6 nnex 11635 . . . . . . 7 ℕ ∈ V
7 ovex 7172 . . . . . . 7 (𝑅𝑟𝑘) ∈ V
86, 7iunex 7655 . . . . . 6 𝑘 ∈ ℕ (𝑅𝑟𝑘) ∈ V
94, 5, 8fvmpt 6749 . . . . 5 (𝑅 ∈ V → (t+‘𝑅) = 𝑘 ∈ ℕ (𝑅𝑟𝑘))
109imaeq1d 5899 . . . 4 (𝑅 ∈ V → ((t+‘𝑅) “ 𝐴) = ( 𝑘 ∈ ℕ (𝑅𝑟𝑘) “ 𝐴))
11 imaiun1 40339 . . . 4 ( 𝑘 ∈ ℕ (𝑅𝑟𝑘) “ 𝐴) = 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴)
1210, 11eqtrdi 2852 . . 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 7147 . . . . . . . . 9 (𝑥 = 1 → (𝑅𝑟𝑥) = (𝑅𝑟1))
1514imaeq1d 5899 . . . . . . . 8 (𝑥 = 1 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟1) “ 𝐴))
1615sseq1d 3949 . . . . . . 7 (𝑥 = 1 → (((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅𝑟1) “ 𝐴) ⊆ 𝐵))
1716imbi2d 344 . . . . . 6 (𝑥 = 1 → (((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟1) “ 𝐴) ⊆ 𝐵)))
18 oveq2 7147 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑅𝑟𝑥) = (𝑅𝑟𝑦))
1918imaeq1d 5899 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟𝑦) “ 𝐴))
2019sseq1d 3949 . . . . . . 7 (𝑥 = 𝑦 → (((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵))
2120imbi2d 344 . . . . . 6 (𝑥 = 𝑦 → (((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵)))
22 oveq2 7147 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑅𝑟𝑥) = (𝑅𝑟(𝑦 + 1)))
2322imaeq1d 5899 . . . . . . . 8 (𝑥 = (𝑦 + 1) → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟(𝑦 + 1)) “ 𝐴))
2423sseq1d 3949 . . . . . . 7 (𝑥 = (𝑦 + 1) → (((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵))
2524imbi2d 344 . . . . . 6 (𝑥 = (𝑦 + 1) → (((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)))
26 oveq2 7147 . . . . . . . . 9 (𝑥 = 𝑘 → (𝑅𝑟𝑥) = (𝑅𝑟𝑘))
2726imaeq1d 5899 . . . . . . . 8 (𝑥 = 𝑘 → ((𝑅𝑟𝑥) “ 𝐴) = ((𝑅𝑟𝑘) “ 𝐴))
2827sseq1d 3949 . . . . . . 7 (𝑥 = 𝑘 → (((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵 ↔ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵))
2928imbi2d 344 . . . . . 6 (𝑥 = 𝑘 → (((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑥) “ 𝐴) ⊆ 𝐵) ↔ ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)))
30 relexp1g 14380 . . . . . . . . 9 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
3130imaeq1d 5899 . . . . . . . 8 (𝑅𝑉 → ((𝑅𝑟1) “ 𝐴) = (𝑅𝐴))
3231adantr 484 . . . . . . 7 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟1) “ 𝐴) = (𝑅𝐴))
33 ssun1 4102 . . . . . . . . 9 𝐴 ⊆ (𝐴𝐵)
34 imass2 5936 . . . . . . . . 9 (𝐴 ⊆ (𝐴𝐵) → (𝑅𝐴) ⊆ (𝑅 “ (𝐴𝐵)))
3533, 34mp1i 13 . . . . . . . 8 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑅𝐴) ⊆ (𝑅 “ (𝐴𝐵)))
36 simpr 488 . . . . . . . 8 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑅 “ (𝐴𝐵)) ⊆ 𝐵)
3735, 36sstrd 3928 . . . . . . 7 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑅𝐴) ⊆ 𝐵)
3832, 37eqsstrd 3956 . . . . . 6 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟1) “ 𝐴) ⊆ 𝐵)
39 simp2l 1196 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑅𝑉)
40 simp1 1133 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → 𝑦 ∈ ℕ)
41 relexpsucnnl 14386 . . . . . . . . . . . 12 ((𝑅𝑉𝑦 ∈ ℕ) → (𝑅𝑟(𝑦 + 1)) = (𝑅 ∘ (𝑅𝑟𝑦)))
4241imaeq1d 5899 . . . . . . . . . . 11 ((𝑅𝑉𝑦 ∈ ℕ) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = ((𝑅 ∘ (𝑅𝑟𝑦)) “ 𝐴))
43 imaco 6075 . . . . . . . . . . 11 ((𝑅 ∘ (𝑅𝑟𝑦)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴))
4442, 43eqtrdi 2852 . . . . . . . . . 10 ((𝑅𝑉𝑦 ∈ ℕ) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)))
4539, 40, 44syl2anc 587 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) = (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)))
46 imass2 5936 . . . . . . . . . . 11 (((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵 → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ (𝑅𝐵))
47463ad2ant3 1132 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ (𝑅𝐵))
48 ssun2 4103 . . . . . . . . . . . 12 𝐵 ⊆ (𝐴𝐵)
49 imass2 5936 . . . . . . . . . . . 12 (𝐵 ⊆ (𝐴𝐵) → (𝑅𝐵) ⊆ (𝑅 “ (𝐴𝐵)))
5048, 49mp1i 13 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅𝐵) ⊆ (𝑅 “ (𝐴𝐵)))
51 simp2r 1197 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ (𝐴𝐵)) ⊆ 𝐵)
5250, 51sstrd 3928 . . . . . . . . . 10 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅𝐵) ⊆ 𝐵)
5347, 52sstrd 3928 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → (𝑅 “ ((𝑅𝑟𝑦) “ 𝐴)) ⊆ 𝐵)
5445, 53eqsstrd 3956 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) ∧ ((𝑅𝑟𝑦) “ 𝐴) ⊆ 𝐵) → ((𝑅𝑟(𝑦 + 1)) “ 𝐴) ⊆ 𝐵)
55543exp 1116 . . . . . . 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 11647 . . . . 5 (𝑘 ∈ ℕ → ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵))
5857com12 32 . . . 4 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → (𝑘 ∈ ℕ → ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵))
5958ralrimiv 3151 . . 3 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ∀𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)
60 iunss 4935 . . 3 ( 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵 ↔ ∀𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)
6159, 60sylibr 237 . 2 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → 𝑘 ∈ ℕ ((𝑅𝑟𝑘) “ 𝐴) ⊆ 𝐵)
6213, 61eqsstrd 3956 1 ((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  Vcvv 3444  cun 3882  wss 3884   ciun 4884  cima 5526  ccom 5527  cfv 6328  (class class class)co 7139  1c1 10531   + caddc 10533  cn 11629  t+ctcl 14340  𝑟crelexp 14374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-seq 13369  df-trcl 14342  df-relexp 14375
This theorem is referenced by:  brtrclfv2  40415  frege77d  40434
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