Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trclexi Structured version   Visualization version   GIF version

Theorem trclexi 43633
Description: The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
Hypothesis
Ref Expression
trclexi.1 𝐴𝑉
Assertion
Ref Expression
trclexi {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem trclexi
StepHypRef Expression
1 ssun1 4178 . 2 𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2 coundir 6268 . . . 4 ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))))
3 coundi 6267 . . . . . 6 (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴)))
4 cossxp 6292 . . . . . . 7 (𝐴𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5 cossxp 6292 . . . . . . . 8 (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom (dom 𝐴 × ran 𝐴) × ran 𝐴)
6 dmxpss 6191 . . . . . . . . 9 dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴
7 xpss1 5704 . . . . . . . . 9 (dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴 → (dom (dom 𝐴 × ran 𝐴) × ran 𝐴) ⊆ (dom 𝐴 × ran 𝐴))
86, 7ax-mp 5 . . . . . . . 8 (dom (dom 𝐴 × ran 𝐴) × ran 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
95, 8sstri 3993 . . . . . . 7 (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
104, 9unssi 4191 . . . . . 6 ((𝐴𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
113, 10eqsstri 4030 . . . . 5 (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
12 coundi 6267 . . . . . 6 ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)))
13 cossxp 6292 . . . . . . . 8 ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran (dom 𝐴 × ran 𝐴))
14 rnxpss 6192 . . . . . . . . 9 ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴
15 xpss2 5705 . . . . . . . . 9 (ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴 → (dom 𝐴 × ran (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴))
1614, 15ax-mp 5 . . . . . . . 8 (dom 𝐴 × ran (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
1713, 16sstri 3993 . . . . . . 7 ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
18 xptrrel 15019 . . . . . . 7 ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
1917, 18unssi 4191 . . . . . 6 (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
2012, 19eqsstri 4030 . . . . 5 ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
2111, 20unssi 4191 . . . 4 ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))) ⊆ (dom 𝐴 × ran 𝐴)
222, 21eqsstri 4030 . . 3 ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
23 ssun2 4179 . . 3 (dom 𝐴 × ran 𝐴) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2422, 23sstri 3993 . 2 ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
25 trclexi.1 . . . . . 6 𝐴𝑉
2625elexi 3503 . . . . 5 𝐴 ∈ V
2726dmex 7931 . . . . . 6 dom 𝐴 ∈ V
2826rnex 7932 . . . . . 6 ran 𝐴 ∈ V
2927, 28xpex 7773 . . . . 5 (dom 𝐴 × ran 𝐴) ∈ V
3026, 29unex 7764 . . . 4 (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∈ V
31 trcleq2lem 15030 . . . 4 (𝑥 = (𝐴 ∪ (dom 𝐴 × ran 𝐴)) → ((𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ (𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∧ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))))
3230, 31spcev 3606 . . 3 ((𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∧ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) → ∃𝑥(𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥))
33 intexab 5346 . . 3 (∃𝑥(𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V)
3432, 33sylib 218 . 2 ((𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∧ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) → {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V)
351, 24, 34mp2an 692 1 {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1779  wcel 2108  {cab 2714  Vcvv 3480  cun 3949  wss 3951   cint 4946   × cxp 5683  dom cdm 5685  ran crn 5686  ccom 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697
This theorem is referenced by:  dfrtrcl5  43642
  Copyright terms: Public domain W3C validator