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Theorem trclexi 38447
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 3919 . 2 𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2 coundir 5798 . . . 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 5797 . . . . . 6 (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴)))
4 cossxp 5819 . . . . . . 7 (𝐴𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5 cossxp 5819 . . . . . . . 8 (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom (dom 𝐴 × ran 𝐴) × ran 𝐴)
6 dmxpss 5723 . . . . . . . . 9 dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴
7 xpss1 5284 . . . . . . . . 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 3753 . . . . . . 7 (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
104, 9unssi 3931 . . . . . 6 ((𝐴𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
113, 10eqsstri 3776 . . . . 5 (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
12 coundi 5797 . . . . . 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 5819 . . . . . . . 8 ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran (dom 𝐴 × ran 𝐴))
14 rnxpss 5724 . . . . . . . . 9 ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴
15 xpss2 5285 . . . . . . . . 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 3753 . . . . . . 7 ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
18 xptrrel 13940 . . . . . . 7 ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
1917, 18unssi 3931 . . . . . 6 (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
2012, 19eqsstri 3776 . . . . 5 ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
2111, 20unssi 3931 . . . 4 ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))) ⊆ (dom 𝐴 × ran 𝐴)
222, 21eqsstri 3776 . . 3 ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
23 ssun2 3920 . . 3 (dom 𝐴 × ran 𝐴) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2422, 23sstri 3753 . 2 ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
25 trclexi.1 . . . . . 6 𝐴𝑉
2625elexi 3353 . . . . 5 𝐴 ∈ V
2726dmex 7265 . . . . . 6 dom 𝐴 ∈ V
2826rnex 7266 . . . . . 6 ran 𝐴 ∈ V
2927, 28xpex 7128 . . . . 5 (dom 𝐴 × ran 𝐴) ∈ V
3026, 29unex 7122 . . . 4 (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∈ V
31 trcleq2lem 13951 . . . 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 3440 . . 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 4971 . . 3 (∃𝑥(𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V)
3432, 33sylib 208 . 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 710 1 {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 383  wex 1853  wcel 2139  {cab 2746  Vcvv 3340  cun 3713  wss 3715   cint 4627   × cxp 5264  dom cdm 5266  ran crn 5267  ccom 5270
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-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-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278
This theorem is referenced by:  dfrtrcl5  38456
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