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Theorem dfrcl2 38641
Description: Reflexive closure of a relation as union with restricted identity relation. (Contributed by RP, 6-Jun-2020.)
Assertion
Ref Expression
dfrcl2 r* = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))

Proof of Theorem dfrcl2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-rcl 38640 . 2 r* = (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
2 rabab 3376 . . . . . . . 8 {𝑧 ∈ V ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} = {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}
32eqcomi 2774 . . . . . . 7 {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} = {𝑧 ∈ V ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}
43inteqi 4637 . . . . . 6 {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} = {𝑧 ∈ V ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}
54a1i 11 . . . . 5 (𝑥 ∈ V → {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} = {𝑧 ∈ V ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
6 vex 3353 . . . . . . . . . . 11 𝑥 ∈ V
76dmex 7297 . . . . . . . . . 10 dom 𝑥 ∈ V
86rnex 7298 . . . . . . . . . 10 ran 𝑥 ∈ V
97, 8unex 7154 . . . . . . . . 9 (dom 𝑥 ∪ ran 𝑥) ∈ V
10 resiexg 7300 . . . . . . . . 9 ((dom 𝑥 ∪ ran 𝑥) ∈ V → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V)
119, 10ax-mp 5 . . . . . . . 8 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V
1211, 6unex 7154 . . . . . . 7 (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∈ V
1312a1i 11 . . . . . 6 (𝑥 ∈ V → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∈ V)
14 ssun2 3939 . . . . . . 7 𝑥 ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)
15 dmun 5499 . . . . . . . . . . . 12 dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) = (dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ dom 𝑥)
16 dmresi 5641 . . . . . . . . . . . . 13 dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
1716uneq1i 3925 . . . . . . . . . . . 12 (dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ dom 𝑥) = ((dom 𝑥 ∪ ran 𝑥) ∪ dom 𝑥)
18 un23 3934 . . . . . . . . . . . . 13 ((dom 𝑥 ∪ ran 𝑥) ∪ dom 𝑥) = ((dom 𝑥 ∪ dom 𝑥) ∪ ran 𝑥)
19 unidm 3918 . . . . . . . . . . . . . 14 (dom 𝑥 ∪ dom 𝑥) = dom 𝑥
2019uneq1i 3925 . . . . . . . . . . . . 13 ((dom 𝑥 ∪ dom 𝑥) ∪ ran 𝑥) = (dom 𝑥 ∪ ran 𝑥)
2118, 20eqtri 2787 . . . . . . . . . . . 12 ((dom 𝑥 ∪ ran 𝑥) ∪ dom 𝑥) = (dom 𝑥 ∪ ran 𝑥)
2215, 17, 213eqtri 2791 . . . . . . . . . . 11 dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) = (dom 𝑥 ∪ ran 𝑥)
23 rnun 5724 . . . . . . . . . . . 12 ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) = (ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ ran 𝑥)
24 rnresi 5661 . . . . . . . . . . . . 13 ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
2524uneq1i 3925 . . . . . . . . . . . 12 (ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ ran 𝑥) = ((dom 𝑥 ∪ ran 𝑥) ∪ ran 𝑥)
26 unass 3932 . . . . . . . . . . . . 13 ((dom 𝑥 ∪ ran 𝑥) ∪ ran 𝑥) = (dom 𝑥 ∪ (ran 𝑥 ∪ ran 𝑥))
27 unidm 3918 . . . . . . . . . . . . . 14 (ran 𝑥 ∪ ran 𝑥) = ran 𝑥
2827uneq2i 3926 . . . . . . . . . . . . 13 (dom 𝑥 ∪ (ran 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
2926, 28eqtri 2787 . . . . . . . . . . . 12 ((dom 𝑥 ∪ ran 𝑥) ∪ ran 𝑥) = (dom 𝑥 ∪ ran 𝑥)
3023, 25, 293eqtri 2791 . . . . . . . . . . 11 ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) = (dom 𝑥 ∪ ran 𝑥)
3122, 30uneq12i 3927 . . . . . . . . . 10 (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)) = ((dom 𝑥 ∪ ran 𝑥) ∪ (dom 𝑥 ∪ ran 𝑥))
32 unidm 3918 . . . . . . . . . 10 ((dom 𝑥 ∪ ran 𝑥) ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
3331, 32eqtri 2787 . . . . . . . . 9 (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
3433reseq2i 5562 . . . . . . . 8 ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
35 ssun1 3938 . . . . . . . 8 ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)
3634, 35eqsstri 3795 . . . . . . 7 ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)
3714, 36pm3.2i 462 . . . . . 6 (𝑥 ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∧ ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
38 dmeq 5492 . . . . . . . . . . 11 (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → dom 𝑧 = dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
39 rneq 5519 . . . . . . . . . . 11 (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → ran 𝑧 = ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
4038, 39uneq12d 3930 . . . . . . . . . 10 (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → (dom 𝑧 ∪ ran 𝑧) = (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)))
4140reseq2d 5565 . . . . . . . . 9 (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → ( I ↾ (dom 𝑧 ∪ ran 𝑧)) = ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))))
42 id 22 . . . . . . . . 9 (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → 𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
4341, 42sseq12d 3794 . . . . . . . 8 (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → (( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧 ↔ ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)))
4443cleq2lem 38589 . . . . . . 7 (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → ((𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) ↔ (𝑥 ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∧ ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))))
4544intminss 4659 . . . . . 6 (((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∈ V ∧ (𝑥 ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∧ ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) → {𝑧 ∈ V ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
4613, 37, 45sylancl 580 . . . . 5 (𝑥 ∈ V → {𝑧 ∈ V ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
475, 46eqsstrd 3799 . . . 4 (𝑥 ∈ V → {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
48 dmss 5491 . . . . . . . . . . . . . . . 16 (𝑥𝑧 → dom 𝑥 ⊆ dom 𝑧)
49 rnss 5522 . . . . . . . . . . . . . . . 16 (𝑥𝑧 → ran 𝑥 ⊆ ran 𝑧)
50 unss12 3947 . . . . . . . . . . . . . . . 16 ((dom 𝑥 ⊆ dom 𝑧 ∧ ran 𝑥 ⊆ ran 𝑧) → (dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑧 ∪ ran 𝑧))
5148, 49, 50syl2anc 579 . . . . . . . . . . . . . . 15 (𝑥𝑧 → (dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑧 ∪ ran 𝑧))
52 dfss 3747 . . . . . . . . . . . . . . 15 ((dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑧 ∪ ran 𝑧) ↔ (dom 𝑥 ∪ ran 𝑥) = ((dom 𝑥 ∪ ran 𝑥) ∩ (dom 𝑧 ∪ ran 𝑧)))
5351, 52sylib 209 . . . . . . . . . . . . . 14 (𝑥𝑧 → (dom 𝑥 ∪ ran 𝑥) = ((dom 𝑥 ∪ ran 𝑥) ∩ (dom 𝑧 ∪ ran 𝑧)))
54 incom 3967 . . . . . . . . . . . . . 14 ((dom 𝑥 ∪ ran 𝑥) ∩ (dom 𝑧 ∪ ran 𝑧)) = ((dom 𝑧 ∪ ran 𝑧) ∩ (dom 𝑥 ∪ ran 𝑥))
5553, 54syl6eq 2815 . . . . . . . . . . . . 13 (𝑥𝑧 → (dom 𝑥 ∪ ran 𝑥) = ((dom 𝑧 ∪ ran 𝑧) ∩ (dom 𝑥 ∪ ran 𝑥)))
5655reseq2d 5565 . . . . . . . . . . . 12 (𝑥𝑧 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ ((dom 𝑧 ∪ ran 𝑧) ∩ (dom 𝑥 ∪ ran 𝑥))))
57 resres 5585 . . . . . . . . . . . 12 (( I ↾ (dom 𝑧 ∪ ran 𝑧)) ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ ((dom 𝑧 ∪ ran 𝑧) ∩ (dom 𝑥 ∪ ran 𝑥)))
5856, 57syl6eqr 2817 . . . . . . . . . . 11 (𝑥𝑧 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (( I ↾ (dom 𝑧 ∪ ran 𝑧)) ↾ (dom 𝑥 ∪ ran 𝑥)))
59 resss 5597 . . . . . . . . . . . 12 (( I ↾ (dom 𝑧 ∪ ran 𝑧)) ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑧 ∪ ran 𝑧))
6059a1i 11 . . . . . . . . . . 11 (𝑥𝑧 → (( I ↾ (dom 𝑧 ∪ ran 𝑧)) ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑧 ∪ ran 𝑧)))
6158, 60eqsstrd 3799 . . . . . . . . . 10 (𝑥𝑧 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑧 ∪ ran 𝑧)))
6261adantr 472 . . . . . . . . 9 ((𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑧 ∪ ran 𝑧)))
63 simpr 477 . . . . . . . . 9 ((𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)
6462, 63sstrd 3771 . . . . . . . 8 ((𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧)
65 simpl 474 . . . . . . . 8 ((𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → 𝑥𝑧)
6664, 65unssd 3951 . . . . . . 7 ((𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ 𝑧)
6766ax-gen 1890 . . . . . 6 𝑧((𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ 𝑧)
6867a1i 11 . . . . 5 (𝑥 ∈ V → ∀𝑧((𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ 𝑧))
69 ssintab 4650 . . . . 5 ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} ↔ ∀𝑧((𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ 𝑧))
7068, 69sylibr 225 . . . 4 (𝑥 ∈ V → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
7147, 70eqssd 3778 . . 3 (𝑥 ∈ V → {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
7271mpteq2ia 4899 . 2 (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}) = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
731, 72eqtri 2787 1 r* = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1650   = wceq 1652  wcel 2155  {cab 2751  {crab 3059  Vcvv 3350  cun 3730  cin 3731  wss 3732   cint 4633  cmpt 4888   I cid 5184  dom cdm 5277  ran crn 5278  cres 5279  r*crcl 38639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-rcl 38640
This theorem is referenced by:  dfrcl3  38642
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