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Theorem iunrelexpmin1 36813
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 788 . . . . 5 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → 𝑁 = ℕ)
4 simpr 476 . . . . . 6 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
54oveq1d 6542 . . . . 5 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
63, 5iuneq12d 4477 . . . 4 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → 𝑛𝑁 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ (𝑅𝑟𝑛))
7 elex 3185 . . . . 5 (𝑅𝑉𝑅 ∈ V)
87adantr 480 . . . 4 ((𝑅𝑉𝑁 = ℕ) → 𝑅 ∈ V)
9 nnex 10876 . . . . . 6 ℕ ∈ V
10 ovex 6555 . . . . . 6 (𝑅𝑟𝑛) ∈ V
119, 10iunex 7017 . . . . 5 𝑛 ∈ ℕ (𝑅𝑟𝑛) ∈ V
1211a1i 11 . . . 4 ((𝑅𝑉𝑁 = ℕ) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ∈ V)
132, 6, 8, 12fvmptd 6182 . . 3 ((𝑅𝑉𝑁 = ℕ) → (𝐶𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛))
14 relexp1g 13563 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
1514sseq1d 3595 . . . . . . 7 (𝑅𝑉 → ((𝑅𝑟1) ⊆ 𝑠𝑅𝑠))
1615anbi1d 737 . . . . . 6 (𝑅𝑉 → (((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) ↔ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
17 oveq2 6535 . . . . . . . . . . . . 13 (𝑥 = 1 → (𝑅𝑟𝑥) = (𝑅𝑟1))
1817sseq1d 3595 . . . . . . . . . . . 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 6535 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑅𝑟𝑥) = (𝑅𝑟𝑦))
2120sseq1d 3595 . . . . . . . . . . . 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 6535 . . . . . . . . . . . . 13 (𝑥 = (𝑦 + 1) → (𝑅𝑟𝑥) = (𝑅𝑟(𝑦 + 1)))
2423sseq1d 3595 . . . . . . . . . . . 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 6535 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (𝑅𝑟𝑥) = (𝑅𝑟𝑛))
2726sseq1d 3595 . . . . . . . . . . . 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 790 . . . . . . . . . . 11 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟1) ⊆ 𝑠)
30 simp1 1054 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → 𝑦 ∈ ℕ)
31 1nn 10881 . . . . . . . . . . . . . . . 16 1 ∈ ℕ
3231a1i 11 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → 1 ∈ ℕ)
33 simp2l 1080 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → 𝑅𝑉)
34 relexpaddnn 13588 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 1 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑦) ∘ (𝑅𝑟1)) = (𝑅𝑟(𝑦 + 1)))
3530, 32, 33, 34syl3anc 1318 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → ((𝑅𝑟𝑦) ∘ (𝑅𝑟1)) = (𝑅𝑟(𝑦 + 1)))
36 simp2rr 1124 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑠𝑠) ⊆ 𝑠)
37 simp3 1056 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑅𝑟𝑦) ⊆ 𝑠)
38 simp2rl 1123 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑅𝑟1) ⊆ 𝑠)
3936, 37, 38trrelssd 13509 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → ((𝑅𝑟𝑦) ∘ (𝑅𝑟1)) ⊆ 𝑠)
4035, 39eqsstr3d 3603 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑅𝑟(𝑦 + 1)) ⊆ 𝑠)
41403exp 1256 . . . . . . . . . . . 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 10888 . . . . . . . . . 10 (𝑛 ∈ ℕ → ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑛) ⊆ 𝑠))
4443com12 32 . . . . . . . . 9 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑛 ∈ ℕ → (𝑅𝑟𝑛) ⊆ 𝑠))
4544ralrimiv 2948 . . . . . . . 8 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → ∀𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠)
46 iunss 4492 . . . . . . . 8 ( 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠 ↔ ∀𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠)
4745, 46sylibr 223 . . . . . . 7 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠)
4847ex 449 . . . . . 6 (𝑅𝑉 → (((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠))
4916, 48sylbird 249 . . . . 5 (𝑅𝑉 → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠))
5049adantr 480 . . . 4 ((𝑅𝑉𝑁 = ℕ) → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠))
51 sseq1 3589 . . . . 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 235 . . 3 ((𝐶𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛) → ((𝑅𝑉𝑁 = ℕ) → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠)))
5413, 53mpcom 37 . 2 ((𝑅𝑉𝑁 = ℕ) → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))
5554alrimiv 1842 1 ((𝑅𝑉𝑁 = ℕ) → ∀𝑠((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031  wal 1473   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  wss 3540   ciun 4450  cmpt 4638  ccom 5032  cfv 5790  (class class class)co 6527  1c1 9794   + caddc 9796  cn 10870  𝑟crelexp 13557
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 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
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 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-n0 11143  df-z 11214  df-uz 11523  df-seq 12622  df-relexp 13558
This theorem is referenced by:  dftrcl3  36825
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