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Theorem trclexi 40693
 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 4077 . 2 𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2 coundir 6078 . . . 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 6077 . . . . . 6 (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴)))
4 cossxp 6101 . . . . . . 7 (𝐴𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5 cossxp 6101 . . . . . . . 8 (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom (dom 𝐴 × ran 𝐴) × ran 𝐴)
6 dmxpss 6000 . . . . . . . . 9 dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴
7 xpss1 5543 . . . . . . . . 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 3901 . . . . . . 7 (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
104, 9unssi 4090 . . . . . 6 ((𝐴𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
113, 10eqsstri 3926 . . . . 5 (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
12 coundi 6077 . . . . . 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 6101 . . . . . . . 8 ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran (dom 𝐴 × ran 𝐴))
14 rnxpss 6001 . . . . . . . . 9 ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴
15 xpss2 5544 . . . . . . . . 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 3901 . . . . . . 7 ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
18 xptrrel 14387 . . . . . . 7 ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
1917, 18unssi 4090 . . . . . 6 (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
2012, 19eqsstri 3926 . . . . 5 ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
2111, 20unssi 4090 . . . 4 ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))) ⊆ (dom 𝐴 × ran 𝐴)
222, 21eqsstri 3926 . . 3 ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
23 ssun2 4078 . . 3 (dom 𝐴 × ran 𝐴) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2422, 23sstri 3901 . 2 ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
25 trclexi.1 . . . . . 6 𝐴𝑉
2625elexi 3429 . . . . 5 𝐴 ∈ V
2726dmex 7621 . . . . . 6 dom 𝐴 ∈ V
2826rnex 7622 . . . . . 6 ran 𝐴 ∈ V
2927, 28xpex 7474 . . . . 5 (dom 𝐴 × ran 𝐴) ∈ V
3026, 29unex 7467 . . . 4 (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∈ V
31 trcleq2lem 14398 . . . 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 3525 . . 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 5209 . . 3 (∃𝑥(𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V)
3432, 33sylib 221 . 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 399  ∃wex 1781   ∈ wcel 2111  {cab 2735  Vcvv 3409   ∪ cun 3856   ⊆ wss 3858  ∩ cint 4838   × cxp 5522  dom cdm 5524  ran crn 5525   ∘ ccom 5528 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-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-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536 This theorem is referenced by:  dfrtrcl5  40702
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