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Theorem undmrnresiss 43176
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 43177. (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 5999 . . 3 ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran 𝐴))
21sseq1i 4005 . 2 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ (( I ↾ dom 𝐴) ∪ ( I ↾ ran 𝐴)) ⊆ 𝐵)
3 unss 4182 . 2 ((( I ↾ dom 𝐴) ⊆ 𝐵 ∧ ( I ↾ ran 𝐴) ⊆ 𝐵) ↔ (( I ↾ dom 𝐴) ∪ ( I ↾ ran 𝐴)) ⊆ 𝐵)
4 relres 6011 . . . . . 6 Rel ( I ↾ dom 𝐴)
5 ssrel 5784 . . . . . 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 vex 3465 . . . . . . . . . 10 𝑥 ∈ V
87eldm 5903 . . . . . . . . 9 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦)
9 df-br 5150 . . . . . . . . . 10 (𝑥 I 𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ I )
10 vex 3465 . . . . . . . . . . 11 𝑧 ∈ V
1110ideq 5855 . . . . . . . . . 10 (𝑥 I 𝑧𝑥 = 𝑧)
129, 11bitr3i 276 . . . . . . . . 9 (⟨𝑥, 𝑧⟩ ∈ I ↔ 𝑥 = 𝑧)
138, 12anbi12ci 627 . . . . . . . 8 ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ I ) ↔ (𝑥 = 𝑧 ∧ ∃𝑦 𝑥𝐴𝑦))
1410opelresi 5993 . . . . . . . 8 (⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ I ))
15 19.42v 1949 . . . . . . . 8 (∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) ↔ (𝑥 = 𝑧 ∧ ∃𝑦 𝑥𝐴𝑦))
1613, 14, 153bitr4i 302 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) ↔ ∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦))
17 df-br 5150 . . . . . . . 8 (𝑥𝐵𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐵)
1817bicomi 223 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ 𝐵𝑥𝐵𝑧)
1916, 18imbi12i 349 . . . . . 6 ((⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝐵) ↔ (∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
20192albii 1814 . . . . 5 (∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) → ⟨𝑥, 𝑧⟩ ∈ 𝐵) ↔ ∀𝑥𝑧(∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
21 19.23v 1937 . . . . . . . 8 (∀𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ (∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
2221bicomi 223 . . . . . . 7 ((∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
23222albii 1814 . . . . . 6 (∀𝑥𝑧(∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑥𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
24 alcom 2148 . . . . . . . 8 (∀𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
25 ancomst 463 . . . . . . . . . . . 12 (((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ((𝑥𝐴𝑦𝑥 = 𝑧) → 𝑥𝐵𝑧))
26 impexp 449 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝑥 = 𝑧) → 𝑥𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)))
2725, 26bitri 274 . . . . . . . . . . 11 (((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)))
2827albii 1813 . . . . . . . . . 10 (∀𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑧(𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)))
29 19.21v 1934 . . . . . . . . . 10 (∀𝑧(𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)) ↔ (𝑥𝐴𝑦 → ∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧)))
30 equcom 2013 . . . . . . . . . . . . . 14 (𝑥 = 𝑧𝑧 = 𝑥)
3130imbi1i 348 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑥𝐵𝑧) ↔ (𝑧 = 𝑥𝑥𝐵𝑧))
3231albii 1813 . . . . . . . . . . . 12 (∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧) ↔ ∀𝑧(𝑧 = 𝑥𝑥𝐵𝑧))
33 breq2 5153 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (𝑥𝐵𝑧𝑥𝐵𝑥))
3433equsalvw 1999 . . . . . . . . . . . 12 (∀𝑧(𝑧 = 𝑥𝑥𝐵𝑧) ↔ 𝑥𝐵𝑥)
3532, 34bitri 274 . . . . . . . . . . 11 (∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧) ↔ 𝑥𝐵𝑥)
3635imbi2i 335 . . . . . . . . . 10 ((𝑥𝐴𝑦 → ∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧)) ↔ (𝑥𝐴𝑦𝑥𝐵𝑥))
3728, 29, 363bitri 296 . . . . . . . . 9 (∀𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ (𝑥𝐴𝑦𝑥𝐵𝑥))
3837albii 1813 . . . . . . . 8 (∀𝑦𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
3924, 38bitri 274 . . . . . . 7 (∀𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
4039albii 1813 . . . . . 6 (∀𝑥𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
4123, 40bitri 274 . . . . 5 (∀𝑥𝑧(∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
426, 20, 413bitri 296 . . . 4 (( I ↾ dom 𝐴) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
43 relres 6011 . . . . . 6 Rel ( I ↾ ran 𝐴)
44 ssrel 5784 . . . . . 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 vex 3465 . . . . . . . . . 10 𝑦 ∈ V
4746elrn 5896 . . . . . . . . 9 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦)
48 df-br 5150 . . . . . . . . . 10 (𝑦 I 𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ I )
4910ideq 5855 . . . . . . . . . 10 (𝑦 I 𝑧𝑦 = 𝑧)
5048, 49bitr3i 276 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ ∈ I ↔ 𝑦 = 𝑧)
5147, 50anbi12ci 627 . . . . . . . 8 ((𝑦 ∈ ran 𝐴 ∧ ⟨𝑦, 𝑧⟩ ∈ I ) ↔ (𝑦 = 𝑧 ∧ ∃𝑥 𝑥𝐴𝑦))
5210opelresi 5993 . . . . . . . 8 (⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) ↔ (𝑦 ∈ ran 𝐴 ∧ ⟨𝑦, 𝑧⟩ ∈ I ))
53 19.42v 1949 . . . . . . . 8 (∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) ↔ (𝑦 = 𝑧 ∧ ∃𝑥 𝑥𝐴𝑦))
5451, 52, 533bitr4i 302 . . . . . . 7 (⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) ↔ ∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦))
55 df-br 5150 . . . . . . . 8 (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
5655bicomi 223 . . . . . . 7 (⟨𝑦, 𝑧⟩ ∈ 𝐵𝑦𝐵𝑧)
5754, 56imbi12i 349 . . . . . 6 ((⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) → ⟨𝑦, 𝑧⟩ ∈ 𝐵) ↔ (∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
58572albii 1814 . . . . 5 (∀𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) → ⟨𝑦, 𝑧⟩ ∈ 𝐵) ↔ ∀𝑦𝑧(∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
59 19.23v 1937 . . . . . . . 8 (∀𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ (∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
6059bicomi 223 . . . . . . 7 ((∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
61602albii 1814 . . . . . 6 (∀𝑦𝑧(∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑦𝑧𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
62 alrot3 2149 . . . . . 6 (∀𝑥𝑦𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑦𝑧𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
63 ancomst 463 . . . . . . . . . 10 (((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ((𝑥𝐴𝑦𝑦 = 𝑧) → 𝑦𝐵𝑧))
64 impexp 449 . . . . . . . . . 10 (((𝑥𝐴𝑦𝑦 = 𝑧) → 𝑦𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)))
6563, 64bitri 274 . . . . . . . . 9 (((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)))
6665albii 1813 . . . . . . . 8 (∀𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑧(𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)))
67 19.21v 1934 . . . . . . . 8 (∀𝑧(𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)) ↔ (𝑥𝐴𝑦 → ∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧)))
68 equcom 2013 . . . . . . . . . . . 12 (𝑦 = 𝑧𝑧 = 𝑦)
6968imbi1i 348 . . . . . . . . . . 11 ((𝑦 = 𝑧𝑦𝐵𝑧) ↔ (𝑧 = 𝑦𝑦𝐵𝑧))
7069albii 1813 . . . . . . . . . 10 (∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧) ↔ ∀𝑧(𝑧 = 𝑦𝑦𝐵𝑧))
71 breq2 5153 . . . . . . . . . . 11 (𝑧 = 𝑦 → (𝑦𝐵𝑧𝑦𝐵𝑦))
7271equsalvw 1999 . . . . . . . . . 10 (∀𝑧(𝑧 = 𝑦𝑦𝐵𝑧) ↔ 𝑦𝐵𝑦)
7370, 72bitri 274 . . . . . . . . 9 (∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧) ↔ 𝑦𝐵𝑦)
7473imbi2i 335 . . . . . . . 8 ((𝑥𝐴𝑦 → ∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧)) ↔ (𝑥𝐴𝑦𝑦𝐵𝑦))
7566, 67, 743bitri 296 . . . . . . 7 (∀𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ (𝑥𝐴𝑦𝑦𝐵𝑦))
76752albii 1814 . . . . . 6 (∀𝑥𝑦𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦))
7761, 62, 763bitr2i 298 . . . . 5 (∀𝑦𝑧(∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦))
7845, 58, 773bitri 296 . . . 4 (( I ↾ ran 𝐴) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦))
7942, 78anbi12i 626 . . 3 ((( I ↾ dom 𝐴) ⊆ 𝐵 ∧ ( I ↾ ran 𝐴) ⊆ 𝐵) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥) ∧ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦)))
80 19.26-2 1866 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝑥𝐵𝑥) ∧ (𝑥𝐴𝑦𝑦𝐵𝑦)) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥) ∧ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦)))
81 pm4.76 517 . . . 4 (((𝑥𝐴𝑦𝑥𝐵𝑥) ∧ (𝑥𝐴𝑦𝑦𝐵𝑦)) ↔ (𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
82812albii 1814 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝑥𝐵𝑥) ∧ (𝑥𝐴𝑦𝑦𝐵𝑦)) ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
8379, 80, 823bitr2i 298 . 2 ((( I ↾ dom 𝐴) ⊆ 𝐵 ∧ ( I ↾ ran 𝐴) ⊆ 𝐵) ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
842, 3, 833bitr2i 298 1 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  wex 1773  wcel 2098  cun 3942  wss 3944  cop 4636   class class class wbr 5149   I cid 5575  dom cdm 5678  ran crn 5679  cres 5680  Rel wrel 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690
This theorem is referenced by:  reflexg  43177
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