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Theorem undmrnresiss 37729
Description: Two ways of saying the identity relation restricted to the union of the domain and range of a relation is a subset of a relation. Generalization of reflexg 37730. (Contributed by RP, 26-Sep-2020.)
Assertion
Ref Expression
undmrnresiss (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem undmrnresiss
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 resundi 5398 . . 3 ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran 𝐴))
21sseq1i 3621 . 2 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ (( I ↾ dom 𝐴) ∪ ( I ↾ ran 𝐴)) ⊆ 𝐵)
3 unss 3779 . 2 ((( I ↾ dom 𝐴) ⊆ 𝐵 ∧ ( I ↾ ran 𝐴) ⊆ 𝐵) ↔ (( I ↾ dom 𝐴) ∪ ( I ↾ ran 𝐴)) ⊆ 𝐵)
4 relres 5414 . . . . . 6 Rel ( I ↾ dom 𝐴)
5 ssrel 5197 . . . . . 6 (Rel ( I ↾ dom 𝐴) → (( I ↾ dom 𝐴) ⊆ 𝐵 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝐵)))
64, 5ax-mp 5 . . . . 5 (( I ↾ dom 𝐴) ⊆ 𝐵 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝐵))
7 df-br 4645 . . . . . . . . . 10 (𝑥 I 𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ I )
8 vex 3198 . . . . . . . . . . 11 𝑧 ∈ V
98ideq 5263 . . . . . . . . . 10 (𝑥 I 𝑧𝑥 = 𝑧)
107, 9bitr3i 266 . . . . . . . . 9 (⟨𝑥, 𝑧⟩ ∈ I ↔ 𝑥 = 𝑧)
11 vex 3198 . . . . . . . . . 10 𝑥 ∈ V
1211eldm 5310 . . . . . . . . 9 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦)
1310, 12anbi12i 732 . . . . . . . 8 ((⟨𝑥, 𝑧⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴) ↔ (𝑥 = 𝑧 ∧ ∃𝑦 𝑥𝐴𝑦))
148opelres 5390 . . . . . . . 8 (⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) ↔ (⟨𝑥, 𝑧⟩ ∈ I ∧ 𝑥 ∈ dom 𝐴))
15 19.42v 1916 . . . . . . . 8 (∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) ↔ (𝑥 = 𝑧 ∧ ∃𝑦 𝑥𝐴𝑦))
1613, 14, 153bitr4i 292 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) ↔ ∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦))
17 df-br 4645 . . . . . . . 8 (𝑥𝐵𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐵)
1817bicomi 214 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ 𝐵𝑥𝐵𝑧)
1916, 18imbi12i 340 . . . . . 6 ((⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝐵) ↔ (∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
20192albii 1746 . . . . 5 (∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝐵) ↔ ∀𝑥𝑧(∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
21 19.23v 1900 . . . . . . . 8 (∀𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ (∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
2221bicomi 214 . . . . . . 7 ((∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
23222albii 1746 . . . . . 6 (∀𝑥𝑧(∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑥𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
24 alcom 2035 . . . . . . . 8 (∀𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
25 ancomst 468 . . . . . . . . . . . 12 (((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ((𝑥𝐴𝑦𝑥 = 𝑧) → 𝑥𝐵𝑧))
26 impexp 462 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝑥 = 𝑧) → 𝑥𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)))
2725, 26bitri 264 . . . . . . . . . . 11 (((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)))
2827albii 1745 . . . . . . . . . 10 (∀𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑧(𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)))
29 19.21v 1866 . . . . . . . . . 10 (∀𝑧(𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)) ↔ (𝑥𝐴𝑦 → ∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧)))
30 equcom 1943 . . . . . . . . . . . . . 14 (𝑥 = 𝑧𝑧 = 𝑥)
3130imbi1i 339 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑥𝐵𝑧) ↔ (𝑧 = 𝑥𝑥𝐵𝑧))
3231albii 1745 . . . . . . . . . . . 12 (∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧) ↔ ∀𝑧(𝑧 = 𝑥𝑥𝐵𝑧))
33 breq2 4648 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (𝑥𝐵𝑧𝑥𝐵𝑥))
3433equsalvw 1929 . . . . . . . . . . . 12 (∀𝑧(𝑧 = 𝑥𝑥𝐵𝑧) ↔ 𝑥𝐵𝑥)
3532, 34bitri 264 . . . . . . . . . . 11 (∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧) ↔ 𝑥𝐵𝑥)
3635imbi2i 326 . . . . . . . . . 10 ((𝑥𝐴𝑦 → ∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧)) ↔ (𝑥𝐴𝑦𝑥𝐵𝑥))
3728, 29, 363bitri 286 . . . . . . . . 9 (∀𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ (𝑥𝐴𝑦𝑥𝐵𝑥))
3837albii 1745 . . . . . . . 8 (∀𝑦𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
3924, 38bitri 264 . . . . . . 7 (∀𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
4039albii 1745 . . . . . 6 (∀𝑥𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
4123, 40bitri 264 . . . . 5 (∀𝑥𝑧(∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
426, 20, 413bitri 286 . . . 4 (( I ↾ dom 𝐴) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
43 relres 5414 . . . . . 6 Rel ( I ↾ ran 𝐴)
44 ssrel 5197 . . . . . 6 (Rel ( I ↾ ran 𝐴) → (( I ↾ ran 𝐴) ⊆ 𝐵 ↔ ∀𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) → ⟨𝑦, 𝑧⟩ ∈ 𝐵)))
4543, 44ax-mp 5 . . . . 5 (( I ↾ ran 𝐴) ⊆ 𝐵 ↔ ∀𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) → ⟨𝑦, 𝑧⟩ ∈ 𝐵))
46 df-br 4645 . . . . . . . . . 10 (𝑦 I 𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ I )
478ideq 5263 . . . . . . . . . 10 (𝑦 I 𝑧𝑦 = 𝑧)
4846, 47bitr3i 266 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ ∈ I ↔ 𝑦 = 𝑧)
49 vex 3198 . . . . . . . . . 10 𝑦 ∈ V
5049elrn 5355 . . . . . . . . 9 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦)
5148, 50anbi12i 732 . . . . . . . 8 ((⟨𝑦, 𝑧⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴) ↔ (𝑦 = 𝑧 ∧ ∃𝑥 𝑥𝐴𝑦))
528opelres 5390 . . . . . . . 8 (⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) ↔ (⟨𝑦, 𝑧⟩ ∈ I ∧ 𝑦 ∈ ran 𝐴))
53 19.42v 1916 . . . . . . . 8 (∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) ↔ (𝑦 = 𝑧 ∧ ∃𝑥 𝑥𝐴𝑦))
5451, 52, 533bitr4i 292 . . . . . . 7 (⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) ↔ ∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦))
55 df-br 4645 . . . . . . . 8 (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
5655bicomi 214 . . . . . . 7 (⟨𝑦, 𝑧⟩ ∈ 𝐵𝑦𝐵𝑧)
5754, 56imbi12i 340 . . . . . 6 ((⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) → ⟨𝑦, 𝑧⟩ ∈ 𝐵) ↔ (∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
58572albii 1746 . . . . 5 (∀𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) → ⟨𝑦, 𝑧⟩ ∈ 𝐵) ↔ ∀𝑦𝑧(∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
59 19.23v 1900 . . . . . . . 8 (∀𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ (∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
6059bicomi 214 . . . . . . 7 ((∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
61602albii 1746 . . . . . 6 (∀𝑦𝑧(∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑦𝑧𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
62 alrot3 2036 . . . . . 6 (∀𝑥𝑦𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑦𝑧𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
63 ancomst 468 . . . . . . . . . 10 (((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ((𝑥𝐴𝑦𝑦 = 𝑧) → 𝑦𝐵𝑧))
64 impexp 462 . . . . . . . . . 10 (((𝑥𝐴𝑦𝑦 = 𝑧) → 𝑦𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)))
6563, 64bitri 264 . . . . . . . . 9 (((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)))
6665albii 1745 . . . . . . . 8 (∀𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑧(𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)))
67 19.21v 1866 . . . . . . . 8 (∀𝑧(𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)) ↔ (𝑥𝐴𝑦 → ∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧)))
68 equcom 1943 . . . . . . . . . . . 12 (𝑦 = 𝑧𝑧 = 𝑦)
6968imbi1i 339 . . . . . . . . . . 11 ((𝑦 = 𝑧𝑦𝐵𝑧) ↔ (𝑧 = 𝑦𝑦𝐵𝑧))
7069albii 1745 . . . . . . . . . 10 (∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧) ↔ ∀𝑧(𝑧 = 𝑦𝑦𝐵𝑧))
71 breq2 4648 . . . . . . . . . . 11 (𝑧 = 𝑦 → (𝑦𝐵𝑧𝑦𝐵𝑦))
7271equsalvw 1929 . . . . . . . . . 10 (∀𝑧(𝑧 = 𝑦𝑦𝐵𝑧) ↔ 𝑦𝐵𝑦)
7370, 72bitri 264 . . . . . . . . 9 (∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧) ↔ 𝑦𝐵𝑦)
7473imbi2i 326 . . . . . . . 8 ((𝑥𝐴𝑦 → ∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧)) ↔ (𝑥𝐴𝑦𝑦𝐵𝑦))
7566, 67, 743bitri 286 . . . . . . 7 (∀𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ (𝑥𝐴𝑦𝑦𝐵𝑦))
76752albii 1746 . . . . . 6 (∀𝑥𝑦𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦))
7761, 62, 763bitr2i 288 . . . . 5 (∀𝑦𝑧(∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦))
7845, 58, 773bitri 286 . . . 4 (( I ↾ ran 𝐴) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦))
7942, 78anbi12i 732 . . 3 ((( I ↾ dom 𝐴) ⊆ 𝐵 ∧ ( I ↾ ran 𝐴) ⊆ 𝐵) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥) ∧ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦)))
80 19.26-2 1797 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝑥𝐵𝑥) ∧ (𝑥𝐴𝑦𝑦𝐵𝑦)) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥) ∧ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦)))
81 pm4.76 909 . . . 4 (((𝑥𝐴𝑦𝑥𝐵𝑥) ∧ (𝑥𝐴𝑦𝑦𝐵𝑦)) ↔ (𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
82812albii 1746 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝑥𝐵𝑥) ∧ (𝑥𝐴𝑦𝑦𝐵𝑦)) ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
8379, 80, 823bitr2i 288 . 2 ((( I ↾ dom 𝐴) ⊆ 𝐵 ∧ ( I ↾ ran 𝐴) ⊆ 𝐵) ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
842, 3, 833bitr2i 288 1 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479  wex 1702  wcel 1988  cun 3565  wss 3567  cop 4174   class class class wbr 4644   I cid 5013  dom cdm 5104  ran crn 5105  cres 5106  Rel wrel 5109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116
This theorem is referenced by:  reflexg  37730
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