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Theorem trclun 13974
 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 3930 . . . . . . . . . 10 ((𝑅𝑥𝑆𝑥) ↔ (𝑅𝑆) ⊆ 𝑥)
2 simpl 474 . . . . . . . . . 10 ((𝑅𝑥𝑆𝑥) → 𝑅𝑥)
31, 2sylbir 225 . . . . . . . . 9 ((𝑅𝑆) ⊆ 𝑥𝑅𝑥)
4 vex 3343 . . . . . . . . . . 11 𝑥 ∈ V
5 trcleq2lem 13951 . . . . . . . . . . 11 (𝑟 = 𝑥 → ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)))
64, 5elab 3490 . . . . . . . . . 10 (𝑥 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ↔ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
76biimpri 218 . . . . . . . . 9 ((𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
83, 7sylan 489 . . . . . . . 8 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
9 intss1 4644 . . . . . . . 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 479 . . . . . . . . . 10 ((𝑅𝑥𝑆𝑥) → 𝑆𝑥)
121, 11sylbir 225 . . . . . . . . 9 ((𝑅𝑆) ⊆ 𝑥𝑆𝑥)
13 trcleq2lem 13951 . . . . . . . . . . 11 (𝑠 = 𝑥 → ((𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) ↔ (𝑆𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)))
144, 13elab 3490 . . . . . . . . . 10 (𝑥 ∈ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ↔ (𝑆𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
1514biimpri 218 . . . . . . . . 9 ((𝑆𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
1612, 15sylan 489 . . . . . . . 8 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → 𝑥 ∈ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
17 intss1 4644 . . . . . . . 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 3932 . . . . . 6 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥)
20 simpr 479 . . . . . 6 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → (𝑥𝑥) ⊆ 𝑥)
2119, 20jca 555 . . . . 5 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
22 ssmin 4648 . . . . . . . 8 𝑅 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
23 ssmin 4648 . . . . . . . 8 𝑆 {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}
24 unss12 3928 . . . . . . . 8 ((𝑅 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∧ 𝑆 {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) → (𝑅𝑆) ⊆ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}))
2522, 23, 24mp2an 710 . . . . . . 7 (𝑅𝑆) ⊆ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
26 sstr 3752 . . . . . . 7 (((𝑅𝑆) ⊆ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ∧ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥) → (𝑅𝑆) ⊆ 𝑥)
2725, 26mpan 708 . . . . . 6 (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 → (𝑅𝑆) ⊆ 𝑥)
2827anim1i 593 . . . . 5 ((( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) → ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
2921, 28impbii 199 . . . 4 (((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
3029abbii 2877 . . 3 {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑥 ∣ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}
3130inteqi 4631 . 2 {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑥 ∣ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}
32 unexg 7125 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅𝑆) ∈ V)
33 trclfv 13960 . . 3 ((𝑅𝑆) ∈ V → (t+‘(𝑅𝑆)) = {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
3432, 33syl 17 . 2 ((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = {𝑥 ∣ ((𝑅𝑆) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
35 simpl 474 . . . . . 6 ((𝑅𝑉𝑆𝑊) → 𝑅𝑉)
36 trclfv 13960 . . . . . 6 (𝑅𝑉 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
3735, 36syl 17 . . . . 5 ((𝑅𝑉𝑆𝑊) → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
38 simpr 479 . . . . . 6 ((𝑅𝑉𝑆𝑊) → 𝑆𝑊)
39 trclfv 13960 . . . . . 6 (𝑆𝑊 → (t+‘𝑆) = {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
4038, 39syl 17 . . . . 5 ((𝑅𝑉𝑆𝑊) → (t+‘𝑆) = {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
4137, 40uneq12d 3911 . . . 4 ((𝑅𝑉𝑆𝑊) → ((t+‘𝑅) ∪ (t+‘𝑆)) = ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}))
4241fveq2d 6357 . . 3 ((𝑅𝑉𝑆𝑊) → (t+‘((t+‘𝑅) ∪ (t+‘𝑆))) = (t+‘( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})))
43 fvex 6363 . . . . . 6 (t+‘𝑅) ∈ V
4436, 43syl6eqelr 2848 . . . . 5 (𝑅𝑉 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∈ V)
45 fvex 6363 . . . . . 6 (t+‘𝑆) ∈ V
4639, 45syl6eqelr 2848 . . . . 5 (𝑆𝑊 {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ∈ V)
47 unexg 7125 . . . . 5 (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∈ V ∧ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ∈ V) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ∈ V)
4844, 46, 47syl2an 495 . . . 4 ((𝑅𝑉𝑆𝑊) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ∈ V)
49 trclfv 13960 . . . 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 2794 . 2 ((𝑅𝑉𝑆𝑊) → (t+‘((t+‘𝑅) ∪ (t+‘𝑆))) = {𝑥 ∣ (( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ∪ {𝑠 ∣ (𝑆𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}) ⊆ 𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
5231, 34, 513eqtr4a 2820 1 ((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  {cab 2746  Vcvv 3340   ∪ cun 3713   ⊆ wss 3715  ∩ cint 4627   ∘ ccom 5270  ‘cfv 6049  t+ctcl 13945 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fv 6057  df-trcl 13947 This theorem is referenced by: (None)
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