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Theorem trclexi 42673
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 4171 . 2 𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2 coundir 6246 . . . 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 6245 . . . . . 6 (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴)))
4 cossxp 6270 . . . . . . 7 (𝐴𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5 cossxp 6270 . . . . . . . 8 (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom (dom 𝐴 × ran 𝐴) × ran 𝐴)
6 dmxpss 6169 . . . . . . . . 9 dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴
7 xpss1 5694 . . . . . . . . 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 3990 . . . . . . 7 (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
104, 9unssi 4184 . . . . . 6 ((𝐴𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
113, 10eqsstri 4015 . . . . 5 (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
12 coundi 6245 . . . . . 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 6270 . . . . . . . 8 ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran (dom 𝐴 × ran 𝐴))
14 rnxpss 6170 . . . . . . . . 9 ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴
15 xpss2 5695 . . . . . . . . 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 3990 . . . . . . 7 ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
18 xptrrel 14931 . . . . . . 7 ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
1917, 18unssi 4184 . . . . . 6 (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
2012, 19eqsstri 4015 . . . . 5 ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
2111, 20unssi 4184 . . . 4 ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))) ⊆ (dom 𝐴 × ran 𝐴)
222, 21eqsstri 4015 . . 3 ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
23 ssun2 4172 . . 3 (dom 𝐴 × ran 𝐴) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2422, 23sstri 3990 . 2 ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
25 trclexi.1 . . . . . 6 𝐴𝑉
2625elexi 3492 . . . . 5 𝐴 ∈ V
2726dmex 7904 . . . . . 6 dom 𝐴 ∈ V
2826rnex 7905 . . . . . 6 ran 𝐴 ∈ V
2927, 28xpex 7742 . . . . 5 (dom 𝐴 × ran 𝐴) ∈ V
3026, 29unex 7735 . . . 4 (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∈ V
31 trcleq2lem 14942 . . . 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 3595 . . 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 5338 . . 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 688 1 {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 394  wex 1779  wcel 2104  {cab 2707  Vcvv 3472  cun 3945  wss 3947   cint 4949   × cxp 5673  dom cdm 5675  ran crn 5676  ccom 5679
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687
This theorem is referenced by:  dfrtrcl5  42682
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