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Theorem trclun 14364
Description: Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.)
Assertion
Ref Expression
trclun ((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))

Proof of Theorem trclun
Dummy variables 𝑥 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unss 4164 . . . . . . . . . 10 ((𝑅𝑥𝑆𝑥) ↔ (𝑅𝑆) ⊆ 𝑥)
2 simpl 483 . . . . . . . . . 10 ((𝑅𝑥𝑆𝑥) → 𝑅𝑥)
31, 2sylbir 236 . . . . . . . . 9 ((𝑅𝑆) ⊆ 𝑥𝑅𝑥)
4 vex 3503 . . . . . . . . . . 11 𝑥 ∈ V
5 trcleq2lem 14341 . . . . . . . . . . 11 (𝑟 = 𝑥 → ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)))
64, 5elab 3671 . . . . . . . . . 10 (𝑥 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ↔ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
76biimpri 229 . . . . . . . . 9 ((𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
83, 7sylan 580 . . . . . . . 8 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
9 intss1 4889 . . . . . . . 8 (𝑥 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} → {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ 𝑥)
108, 9syl 17 . . . . . . 7 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ 𝑥)
11 simpr 485 . . . . . . . . . 10 ((𝑅𝑥𝑆𝑥) → 𝑆𝑥)
121, 11sylbir 236 . . . . . . . . 9 ((𝑅𝑆) ⊆ 𝑥𝑆𝑥)
13 trcleq2lem 14341 . . . . . . . . . . 11 (𝑠 = 𝑥 → ((𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) ↔ (𝑆𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)))
144, 13elab 3671 . . . . . . . . . 10 (𝑥 ∈ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ↔ (𝑆𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
1514biimpri 229 . . . . . . . . 9 ((𝑆𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
1612, 15sylan 580 . . . . . . . 8 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
17 intss1 4889 . . . . . . . 8 (𝑥 ∈ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} → {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ 𝑥)
1816, 17syl 17 . . . . . . 7 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ 𝑥)
1910, 18unssd 4166 . . . . . 6 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥)
20 simpr 485 . . . . . 6 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → (𝑥𝑥) ⊆ 𝑥)
2119, 20jca 512 . . . . 5 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
22 ssmin 4893 . . . . . . . 8 𝑅 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
23 ssmin 4893 . . . . . . . 8 𝑆 {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}
24 unss12 4162 . . . . . . . 8 ((𝑅 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∧ 𝑆 {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) → (𝑅𝑆) ⊆ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}))
2522, 23, 24mp2an 688 . . . . . . 7 (𝑅𝑆) ⊆ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
26 sstr 3979 . . . . . . 7 (((𝑅𝑆) ⊆ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ∧ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥) → (𝑅𝑆) ⊆ 𝑥)
2725, 26mpan 686 . . . . . 6 (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 → (𝑅𝑆) ⊆ 𝑥)
2827anim1i 614 . . . . 5 ((( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
2921, 28impbii 210 . . . 4 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
3029abbii 2891 . . 3 {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑥 ∣ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}
3130inteqi 4878 . 2 {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑥 ∣ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}
32 unexg 7461 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅𝑆) ∈ V)
33 trclfv 14350 . . 3 ((𝑅𝑆) ∈ V → (t+‘(𝑅𝑆)) = {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
3432, 33syl 17 . 2 ((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
35 simpl 483 . . . . . 6 ((𝑅𝑉𝑆𝑊) → 𝑅𝑉)
36 trclfv 14350 . . . . . 6 (𝑅𝑉 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
3735, 36syl 17 . . . . 5 ((𝑅𝑉𝑆𝑊) → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
38 simpr 485 . . . . . 6 ((𝑅𝑉𝑆𝑊) → 𝑆𝑊)
39 trclfv 14350 . . . . . 6 (𝑆𝑊 → (t+‘𝑆) = {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
4038, 39syl 17 . . . . 5 ((𝑅𝑉𝑆𝑊) → (t+‘𝑆) = {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
4137, 40uneq12d 4144 . . . 4 ((𝑅𝑉𝑆𝑊) → ((t+‘𝑅) ∪ (t+‘𝑆)) = ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}))
4241fveq2d 6671 . . 3 ((𝑅𝑉𝑆𝑊) → (t+‘((t+‘𝑅) ∪ (t+‘𝑆))) = (t+‘( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})))
43 fvex 6680 . . . . . 6 (t+‘𝑅) ∈ V
4436, 43syl6eqelr 2927 . . . . 5 (𝑅𝑉 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∈ V)
45 fvex 6680 . . . . . 6 (t+‘𝑆) ∈ V
4639, 45syl6eqelr 2927 . . . . 5 (𝑆𝑊 {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ∈ V)
47 unexg 7461 . . . . 5 (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∈ V ∧ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ∈ V) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ∈ V)
4844, 46, 47syl2an 595 . . . 4 ((𝑅𝑉𝑆𝑊) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ∈ V)
49 trclfv 14350 . . . 4 (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ∈ V → (t+‘( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})) = {𝑥 ∣ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
5048, 49syl 17 . . 3 ((𝑅𝑉𝑆𝑊) → (t+‘( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})) = {𝑥 ∣ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
5142, 50eqtrd 2861 . 2 ((𝑅𝑉𝑆𝑊) → (t+‘((t+‘𝑅) ∪ (t+‘𝑆))) = {𝑥 ∣ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
5231, 34, 513eqtr4a 2887 1 ((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  {cab 2804  Vcvv 3500  cun 3938  wss 3940   cint 4874  ccom 5558  cfv 6352  t+ctcl 14335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-iota 6312  df-fun 6354  df-fv 6360  df-trcl 14337
This theorem is referenced by: (None)
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