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Theorem iunrelexpmin1 38317
Description: The indexed union of relation exponentiation over the natural numbers is the minimum transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
Hypothesis
Ref Expression
iunrelexpmin1.def 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))
Assertion
Ref Expression
iunrelexpmin1 ((𝑅𝑉𝑁 = ℕ) → ∀𝑠((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))
Distinct variable groups:   𝑛,𝑟,𝐶,𝑁   𝑁,𝑠   𝑅,𝑛,𝑟   𝑅,𝑠   𝑛,𝑉,𝑟   𝑉,𝑠,𝑛
Allowed substitution hint:   𝐶(𝑠)

Proof of Theorem iunrelexpmin1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iunrelexpmin1.def . . . . 5 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))
21a1i 11 . . . 4 ((𝑅𝑉𝑁 = ℕ) → 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛)))
3 simplr 807 . . . . 5 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → 𝑁 = ℕ)
4 simpr 476 . . . . . 6 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
54oveq1d 6705 . . . . 5 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
63, 5iuneq12d 4578 . . . 4 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → 𝑛𝑁 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ (𝑅𝑟𝑛))
7 elex 3243 . . . . 5 (𝑅𝑉𝑅 ∈ V)
87adantr 480 . . . 4 ((𝑅𝑉𝑁 = ℕ) → 𝑅 ∈ V)
9 nnex 11064 . . . . . 6 ℕ ∈ V
10 ovex 6718 . . . . . 6 (𝑅𝑟𝑛) ∈ V
119, 10iunex 7189 . . . . 5 𝑛 ∈ ℕ (𝑅𝑟𝑛) ∈ V
1211a1i 11 . . . 4 ((𝑅𝑉𝑁 = ℕ) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ∈ V)
132, 6, 8, 12fvmptd 6327 . . 3 ((𝑅𝑉𝑁 = ℕ) → (𝐶𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛))
14 relexp1g 13810 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
1514sseq1d 3665 . . . . . . 7 (𝑅𝑉 → ((𝑅𝑟1) ⊆ 𝑠𝑅𝑠))
1615anbi1d 741 . . . . . 6 (𝑅𝑉 → (((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) ↔ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
17 oveq2 6698 . . . . . . . . . . . . 13 (𝑥 = 1 → (𝑅𝑟𝑥) = (𝑅𝑟1))
1817sseq1d 3665 . . . . . . . . . . . 12 (𝑥 = 1 → ((𝑅𝑟𝑥) ⊆ 𝑠 ↔ (𝑅𝑟1) ⊆ 𝑠))
1918imbi2d 329 . . . . . . . . . . 11 (𝑥 = 1 → (((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟1) ⊆ 𝑠)))
20 oveq2 6698 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑅𝑟𝑥) = (𝑅𝑟𝑦))
2120sseq1d 3665 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑅𝑟𝑥) ⊆ 𝑠 ↔ (𝑅𝑟𝑦) ⊆ 𝑠))
2221imbi2d 329 . . . . . . . . . . 11 (𝑥 = 𝑦 → (((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑦) ⊆ 𝑠)))
23 oveq2 6698 . . . . . . . . . . . . 13 (𝑥 = (𝑦 + 1) → (𝑅𝑟𝑥) = (𝑅𝑟(𝑦 + 1)))
2423sseq1d 3665 . . . . . . . . . . . 12 (𝑥 = (𝑦 + 1) → ((𝑅𝑟𝑥) ⊆ 𝑠 ↔ (𝑅𝑟(𝑦 + 1)) ⊆ 𝑠))
2524imbi2d 329 . . . . . . . . . . 11 (𝑥 = (𝑦 + 1) → (((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟(𝑦 + 1)) ⊆ 𝑠)))
26 oveq2 6698 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (𝑅𝑟𝑥) = (𝑅𝑟𝑛))
2726sseq1d 3665 . . . . . . . . . . . 12 (𝑥 = 𝑛 → ((𝑅𝑟𝑥) ⊆ 𝑠 ↔ (𝑅𝑟𝑛) ⊆ 𝑠))
2827imbi2d 329 . . . . . . . . . . 11 (𝑥 = 𝑛 → (((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑛) ⊆ 𝑠)))
29 simprl 809 . . . . . . . . . . 11 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟1) ⊆ 𝑠)
30 simp1 1081 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → 𝑦 ∈ ℕ)
31 1nn 11069 . . . . . . . . . . . . . . . 16 1 ∈ ℕ
3231a1i 11 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → 1 ∈ ℕ)
33 simp2l 1107 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → 𝑅𝑉)
34 relexpaddnn 13835 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 1 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑦) ∘ (𝑅𝑟1)) = (𝑅𝑟(𝑦 + 1)))
3530, 32, 33, 34syl3anc 1366 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → ((𝑅𝑟𝑦) ∘ (𝑅𝑟1)) = (𝑅𝑟(𝑦 + 1)))
36 simp2rr 1151 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑠𝑠) ⊆ 𝑠)
37 simp3 1083 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑅𝑟𝑦) ⊆ 𝑠)
38 simp2rl 1150 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑅𝑟1) ⊆ 𝑠)
3936, 37, 38trrelssd 13758 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → ((𝑅𝑟𝑦) ∘ (𝑅𝑟1)) ⊆ 𝑠)
4035, 39eqsstr3d 3673 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑅𝑟(𝑦 + 1)) ⊆ 𝑠)
41403exp 1283 . . . . . . . . . . . 12 (𝑦 ∈ ℕ → ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → ((𝑅𝑟𝑦) ⊆ 𝑠 → (𝑅𝑟(𝑦 + 1)) ⊆ 𝑠)))
4241a2d 29 . . . . . . . . . . 11 (𝑦 ∈ ℕ → (((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑦) ⊆ 𝑠) → ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟(𝑦 + 1)) ⊆ 𝑠)))
4319, 22, 25, 28, 29, 42nnind 11076 . . . . . . . . . 10 (𝑛 ∈ ℕ → ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑛) ⊆ 𝑠))
4443com12 32 . . . . . . . . 9 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑛 ∈ ℕ → (𝑅𝑟𝑛) ⊆ 𝑠))
4544ralrimiv 2994 . . . . . . . 8 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → ∀𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠)
46 iunss 4593 . . . . . . . 8 ( 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠 ↔ ∀𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠)
4745, 46sylibr 224 . . . . . . 7 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠)
4847ex 449 . . . . . 6 (𝑅𝑉 → (((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠))
4916, 48sylbird 250 . . . . 5 (𝑅𝑉 → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠))
5049adantr 480 . . . 4 ((𝑅𝑉𝑁 = ℕ) → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠))
51 sseq1 3659 . . . . 5 ((𝐶𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛) → ((𝐶𝑅) ⊆ 𝑠 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠))
5251imbi2d 329 . . . 4 ((𝐶𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛) → (((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠) ↔ ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠)))
5350, 52syl5ibr 236 . . 3 ((𝐶𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛) → ((𝑅𝑉𝑁 = ℕ) → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠)))
5413, 53mpcom 38 . 2 ((𝑅𝑉𝑁 = ℕ) → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))
5554alrimiv 1895 1 ((𝑅𝑉𝑁 = ℕ) → ∀𝑠((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607   ciun 4552  cmpt 4762  ccom 5147  cfv 5926  (class class class)co 6690  1c1 9975   + caddc 9977  cn 11058  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-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-n0 11331  df-z 11416  df-uz 11726  df-seq 12842  df-relexp 13805
This theorem is referenced by:  dftrcl3  38329
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