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Theorem trclun 14421
 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 4089 . . . . . . . . . 10 ((𝑅𝑥𝑆𝑥) ↔ (𝑅𝑆) ⊆ 𝑥)
2 simpl 486 . . . . . . . . . 10 ((𝑅𝑥𝑆𝑥) → 𝑅𝑥)
31, 2sylbir 238 . . . . . . . . 9 ((𝑅𝑆) ⊆ 𝑥𝑅𝑥)
4 vex 3413 . . . . . . . . . . 11 𝑥 ∈ V
5 trcleq2lem 14398 . . . . . . . . . . 11 (𝑟 = 𝑥 → ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)))
64, 5elab 3588 . . . . . . . . . 10 (𝑥 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ↔ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
76biimpri 231 . . . . . . . . 9 ((𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
83, 7sylan 583 . . . . . . . 8 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
9 intss1 4853 . . . . . . . 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 488 . . . . . . . . . 10 ((𝑅𝑥𝑆𝑥) → 𝑆𝑥)
121, 11sylbir 238 . . . . . . . . 9 ((𝑅𝑆) ⊆ 𝑥𝑆𝑥)
13 trcleq2lem 14398 . . . . . . . . . . 11 (𝑠 = 𝑥 → ((𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) ↔ (𝑆𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)))
144, 13elab 3588 . . . . . . . . . 10 (𝑥 ∈ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ↔ (𝑆𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
1514biimpri 231 . . . . . . . . 9 ((𝑆𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
1612, 15sylan 583 . . . . . . . 8 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
17 intss1 4853 . . . . . . . 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 4091 . . . . . 6 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥)
20 simpr 488 . . . . . 6 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → (𝑥𝑥) ⊆ 𝑥)
2119, 20jca 515 . . . . 5 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
22 ssmin 4857 . . . . . . . 8 𝑅 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
23 ssmin 4857 . . . . . . . 8 𝑆 {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}
24 unss12 4087 . . . . . . . 8 ((𝑅 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∧ 𝑆 {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) → (𝑅𝑆) ⊆ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}))
2522, 23, 24mp2an 691 . . . . . . 7 (𝑅𝑆) ⊆ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
26 sstr 3900 . . . . . . 7 (((𝑅𝑆) ⊆ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ∧ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥) → (𝑅𝑆) ⊆ 𝑥)
2725, 26mpan 689 . . . . . 6 (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 → (𝑅𝑆) ⊆ 𝑥)
2827anim1i 617 . . . . 5 ((( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
2921, 28impbii 212 . . . 4 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
3029abbii 2823 . . 3 {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑥 ∣ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}
3130inteqi 4842 . 2 {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑥 ∣ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}
32 unexg 7470 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅𝑆) ∈ V)
33 trclfv 14407 . . 3 ((𝑅𝑆) ∈ V → (t+‘(𝑅𝑆)) = {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
3432, 33syl 17 . 2 ((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
35 simpl 486 . . . . . 6 ((𝑅𝑉𝑆𝑊) → 𝑅𝑉)
36 trclfv 14407 . . . . . 6 (𝑅𝑉 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
3735, 36syl 17 . . . . 5 ((𝑅𝑉𝑆𝑊) → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
38 simpr 488 . . . . . 6 ((𝑅𝑉𝑆𝑊) → 𝑆𝑊)
39 trclfv 14407 . . . . . 6 (𝑆𝑊 → (t+‘𝑆) = {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
4038, 39syl 17 . . . . 5 ((𝑅𝑉𝑆𝑊) → (t+‘𝑆) = {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
4137, 40uneq12d 4069 . . . 4 ((𝑅𝑉𝑆𝑊) → ((t+‘𝑅) ∪ (t+‘𝑆)) = ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}))
4241fveq2d 6662 . . 3 ((𝑅𝑉𝑆𝑊) → (t+‘((t+‘𝑅) ∪ (t+‘𝑆))) = (t+‘( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})))
43 fvex 6671 . . . . . 6 (t+‘𝑅) ∈ V
4436, 43eqeltrrdi 2861 . . . . 5 (𝑅𝑉 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∈ V)
45 fvex 6671 . . . . . 6 (t+‘𝑆) ∈ V
4639, 45eqeltrrdi 2861 . . . . 5 (𝑆𝑊 {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ∈ V)
47 unexg 7470 . . . . 5 (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∈ V ∧ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ∈ V) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ∈ V)
4844, 46, 47syl2an 598 . . . 4 ((𝑅𝑉𝑆𝑊) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ∈ V)
49 trclfv 14407 . . . 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 2793 . 2 ((𝑅𝑉𝑆𝑊) → (t+‘((t+‘𝑅) ∪ (t+‘𝑆))) = {𝑥 ∣ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
5231, 34, 513eqtr4a 2819 1 ((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  Vcvv 3409   ∪ cun 3856   ⊆ wss 3858  ∩ cint 4838   ∘ ccom 5528  ‘cfv 6335  t+ctcl 14392 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-int 4839  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-iota 6294  df-fun 6337  df-fv 6343  df-trcl 14394 This theorem is referenced by: (None)
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