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Theorem rclexi 38448
Description: The reflexive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
Hypothesis
Ref Expression
rclexi.1 𝐴𝑉
Assertion
Ref Expression
rclexi {𝑥 ∣ (𝐴𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem rclexi
StepHypRef Expression
1 ssun1 3927 . 2 𝐴 ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
2 dmun 5468 . . . . . . 7 dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
3 dmresi 5597 . . . . . . . 8 dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
43uneq2i 3915 . . . . . . 7 (dom 𝐴 ∪ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴))
5 ssun1 3927 . . . . . . . 8 dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 ssequn1 3934 . . . . . . . 8 (dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) ↔ (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴))
75, 6mpbi 220 . . . . . . 7 (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
82, 4, 73eqtri 2797 . . . . . 6 dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ ran 𝐴)
9 rnun 5681 . . . . . . 7 ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
10 rnresi 5619 . . . . . . . 8 ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
1110uneq2i 3915 . . . . . . 7 (ran 𝐴 ∪ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴))
12 ssun2 3928 . . . . . . . 8 ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
13 ssequn1 3934 . . . . . . . 8 (ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) ↔ (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴))
1412, 13mpbi 220 . . . . . . 7 (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
159, 11, 143eqtri 2797 . . . . . 6 ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ ran 𝐴)
168, 15uneq12i 3916 . . . . 5 (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))) = ((dom 𝐴 ∪ ran 𝐴) ∪ (dom 𝐴 ∪ ran 𝐴))
17 unidm 3907 . . . . 5 ((dom 𝐴 ∪ ran 𝐴) ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
1816, 17eqtri 2793 . . . 4 (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))) = (dom 𝐴 ∪ ran 𝐴)
1918reseq2i 5530 . . 3 ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) = ( I ↾ (dom 𝐴 ∪ ran 𝐴))
20 ssun2 3928 . . 3 ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
2119, 20eqsstri 3784 . 2 ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
22 rclexi.1 . . . . . 6 𝐴𝑉
2322elexi 3365 . . . . 5 𝐴 ∈ V
24 dmexg 7247 . . . . . . . 8 (𝐴𝑉 → dom 𝐴 ∈ V)
25 rnexg 7248 . . . . . . . 8 (𝐴𝑉 → ran 𝐴 ∈ V)
26 unexg 7109 . . . . . . . 8 ((dom 𝐴 ∈ V ∧ ran 𝐴 ∈ V) → (dom 𝐴 ∪ ran 𝐴) ∈ V)
2724, 25, 26syl2anc 573 . . . . . . 7 (𝐴𝑉 → (dom 𝐴 ∪ ran 𝐴) ∈ V)
2827resiexd 6626 . . . . . 6 (𝐴𝑉 → ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ∈ V)
2922, 28ax-mp 5 . . . . 5 ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ∈ V
3023, 29unex 7106 . . . 4 (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∈ V
31 dmeq 5461 . . . . . . . 8 (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → dom 𝑥 = dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
32 rneq 5488 . . . . . . . 8 (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → ran 𝑥 = ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
3331, 32uneq12d 3919 . . . . . . 7 (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → (dom 𝑥 ∪ ran 𝑥) = (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))))
3433reseq2d 5533 . . . . . 6 (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))))
35 id 22 . . . . . 6 (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → 𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
3634, 35sseq12d 3783 . . . . 5 (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 ↔ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))))
3736cleq2lem 38440 . . . 4 (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → ((𝐴𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥) ↔ (𝐴 ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))))
3830, 37spcev 3451 . . 3 ((𝐴 ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))) → ∃𝑥(𝐴𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
39 intexab 4954 . . 3 (∃𝑥(𝐴𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥) ↔ {𝑥 ∣ (𝐴𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V)
4038, 39sylib 208 . 2 ((𝐴 ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))) → {𝑥 ∣ (𝐴𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V)
411, 21, 40mp2an 672 1 {𝑥 ∣ (𝐴𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wex 1852  wcel 2145  {cab 2757  Vcvv 3351  cun 3721  wss 3723   cint 4612   I cid 5157  dom cdm 5250  ran crn 5251  cres 5252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038
This theorem is referenced by: (None)
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