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Theorem iunrelexpmin1 39931
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 . . . 4 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))
2 simplr 765 . . . . 5 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → 𝑁 = ℕ)
3 simpr 485 . . . . . 6 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
43oveq1d 7160 . . . . 5 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
52, 4iuneq12d 4938 . . . 4 (((𝑅𝑉𝑁 = ℕ) ∧ 𝑟 = 𝑅) → 𝑛𝑁 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ (𝑅𝑟𝑛))
6 elex 3510 . . . . 5 (𝑅𝑉𝑅 ∈ V)
76adantr 481 . . . 4 ((𝑅𝑉𝑁 = ℕ) → 𝑅 ∈ V)
8 nnex 11632 . . . . . 6 ℕ ∈ V
9 ovex 7178 . . . . . 6 (𝑅𝑟𝑛) ∈ V
108, 9iunex 7658 . . . . 5 𝑛 ∈ ℕ (𝑅𝑟𝑛) ∈ V
1110a1i 11 . . . 4 ((𝑅𝑉𝑁 = ℕ) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ∈ V)
121, 5, 7, 11fvmptd2 6768 . . 3 ((𝑅𝑉𝑁 = ℕ) → (𝐶𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛))
13 relexp1g 14373 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
1413sseq1d 3995 . . . . . . 7 (𝑅𝑉 → ((𝑅𝑟1) ⊆ 𝑠𝑅𝑠))
1514anbi1d 629 . . . . . 6 (𝑅𝑉 → (((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) ↔ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)))
16 oveq2 7153 . . . . . . . . . . . . 13 (𝑥 = 1 → (𝑅𝑟𝑥) = (𝑅𝑟1))
1716sseq1d 3995 . . . . . . . . . . . 12 (𝑥 = 1 → ((𝑅𝑟𝑥) ⊆ 𝑠 ↔ (𝑅𝑟1) ⊆ 𝑠))
1817imbi2d 342 . . . . . . . . . . 11 (𝑥 = 1 → (((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟1) ⊆ 𝑠)))
19 oveq2 7153 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑅𝑟𝑥) = (𝑅𝑟𝑦))
2019sseq1d 3995 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑅𝑟𝑥) ⊆ 𝑠 ↔ (𝑅𝑟𝑦) ⊆ 𝑠))
2120imbi2d 342 . . . . . . . . . . 11 (𝑥 = 𝑦 → (((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑦) ⊆ 𝑠)))
22 oveq2 7153 . . . . . . . . . . . . 13 (𝑥 = (𝑦 + 1) → (𝑅𝑟𝑥) = (𝑅𝑟(𝑦 + 1)))
2322sseq1d 3995 . . . . . . . . . . . 12 (𝑥 = (𝑦 + 1) → ((𝑅𝑟𝑥) ⊆ 𝑠 ↔ (𝑅𝑟(𝑦 + 1)) ⊆ 𝑠))
2423imbi2d 342 . . . . . . . . . . 11 (𝑥 = (𝑦 + 1) → (((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟(𝑦 + 1)) ⊆ 𝑠)))
25 oveq2 7153 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (𝑅𝑟𝑥) = (𝑅𝑟𝑛))
2625sseq1d 3995 . . . . . . . . . . . 12 (𝑥 = 𝑛 → ((𝑅𝑟𝑥) ⊆ 𝑠 ↔ (𝑅𝑟𝑛) ⊆ 𝑠))
2726imbi2d 342 . . . . . . . . . . 11 (𝑥 = 𝑛 → (((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑥) ⊆ 𝑠) ↔ ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑛) ⊆ 𝑠)))
28 simprl 767 . . . . . . . . . . 11 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟1) ⊆ 𝑠)
29 simp1 1128 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → 𝑦 ∈ ℕ)
30 1nn 11637 . . . . . . . . . . . . . . . 16 1 ∈ ℕ
3130a1i 11 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → 1 ∈ ℕ)
32 simp2l 1191 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → 𝑅𝑉)
33 relexpaddnn 14398 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ 1 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑦) ∘ (𝑅𝑟1)) = (𝑅𝑟(𝑦 + 1)))
3429, 31, 32, 33syl3anc 1363 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → ((𝑅𝑟𝑦) ∘ (𝑅𝑟1)) = (𝑅𝑟(𝑦 + 1)))
35 simp2rr 1235 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑠𝑠) ⊆ 𝑠)
36 simp3 1130 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑅𝑟𝑦) ⊆ 𝑠)
37 simp2rl 1234 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑅𝑟1) ⊆ 𝑠)
3835, 36, 37trrelssd 14321 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → ((𝑅𝑟𝑦) ∘ (𝑅𝑟1)) ⊆ 𝑠)
3934, 38eqsstrrd 4003 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕ ∧ (𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) ∧ (𝑅𝑟𝑦) ⊆ 𝑠) → (𝑅𝑟(𝑦 + 1)) ⊆ 𝑠)
40393exp 1111 . . . . . . . . . . . 12 (𝑦 ∈ ℕ → ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → ((𝑅𝑟𝑦) ⊆ 𝑠 → (𝑅𝑟(𝑦 + 1)) ⊆ 𝑠)))
4140a2d 29 . . . . . . . . . . 11 (𝑦 ∈ ℕ → (((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑦) ⊆ 𝑠) → ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟(𝑦 + 1)) ⊆ 𝑠)))
4218, 21, 24, 27, 28, 41nnind 11644 . . . . . . . . . 10 (𝑛 ∈ ℕ → ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑅𝑟𝑛) ⊆ 𝑠))
4342com12 32 . . . . . . . . 9 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → (𝑛 ∈ ℕ → (𝑅𝑟𝑛) ⊆ 𝑠))
4443ralrimiv 3178 . . . . . . . 8 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → ∀𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠)
45 iunss 4960 . . . . . . . 8 ( 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠 ↔ ∀𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠)
4644, 45sylibr 235 . . . . . . 7 ((𝑅𝑉 ∧ ((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠)
4746ex 413 . . . . . 6 (𝑅𝑉 → (((𝑅𝑟1) ⊆ 𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠))
4815, 47sylbird 261 . . . . 5 (𝑅𝑉 → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠))
4948adantr 481 . . . 4 ((𝑅𝑉𝑁 = ℕ) → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠))
50 sseq1 3989 . . . . 5 ((𝐶𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛) → ((𝐶𝑅) ⊆ 𝑠 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠))
5150imbi2d 342 . . . 4 ((𝐶𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛) → (((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠) ↔ ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) ⊆ 𝑠)))
5249, 51syl5ibr 247 . . 3 ((𝐶𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛) → ((𝑅𝑉𝑁 = ℕ) → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠)))
5312, 52mpcom 38 . 2 ((𝑅𝑉𝑁 = ℕ) → ((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))
5453alrimiv 1919 1 ((𝑅𝑉𝑁 = ℕ) → ∀𝑠((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079  wal 1526   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  wss 3933   ciun 4910  cmpt 5137  ccom 5552  cfv 6348  (class class class)co 7145  1c1 10526   + caddc 10528  cn 11626  𝑟crelexp 14367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13358  df-relexp 14368
This theorem is referenced by:  dftrcl3  39943
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