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 42371
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 4173 . 2 𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2 coundir 6248 . . . 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 6247 . . . . . 6 (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴)))
4 cossxp 6272 . . . . . . 7 (𝐴𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5 cossxp 6272 . . . . . . . 8 (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom (dom 𝐴 × ran 𝐴) × ran 𝐴)
6 dmxpss 6171 . . . . . . . . 9 dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴
7 xpss1 5696 . . . . . . . . 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 3992 . . . . . . 7 (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
104, 9unssi 4186 . . . . . 6 ((𝐴𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
113, 10eqsstri 4017 . . . . 5 (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
12 coundi 6247 . . . . . 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 6272 . . . . . . . 8 ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran (dom 𝐴 × ran 𝐴))
14 rnxpss 6172 . . . . . . . . 9 ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴
15 xpss2 5697 . . . . . . . . 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 3992 . . . . . . 7 ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
18 xptrrel 14927 . . . . . . 7 ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
1917, 18unssi 4186 . . . . . 6 (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
2012, 19eqsstri 4017 . . . . 5 ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
2111, 20unssi 4186 . . . 4 ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))) ⊆ (dom 𝐴 × ran 𝐴)
222, 21eqsstri 4017 . . 3 ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
23 ssun2 4174 . . 3 (dom 𝐴 × ran 𝐴) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2422, 23sstri 3992 . 2 ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
25 trclexi.1 . . . . . 6 𝐴𝑉
2625elexi 3494 . . . . 5 𝐴 ∈ V
2726dmex 7902 . . . . . 6 dom 𝐴 ∈ V
2826rnex 7903 . . . . . 6 ran 𝐴 ∈ V
2927, 28xpex 7740 . . . . 5 (dom 𝐴 × ran 𝐴) ∈ V
3026, 29unex 7733 . . . 4 (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∈ V
31 trcleq2lem 14938 . . . 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 3597 . . 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 5340 . . 3 (∃𝑥(𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V)
3432, 33sylib 217 . 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 691 1 {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 397  wex 1782  wcel 2107  {cab 2710  Vcvv 3475  cun 3947  wss 3949   cint 4951   × cxp 5675  dom cdm 5677  ran crn 5678  ccom 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689
This theorem is referenced by:  dfrtrcl5  42380
  Copyright terms: Public domain W3C validator