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Theorem undmrnresiss 39954
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 39955. (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 5860 . . 3 ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran 𝐴))
21sseq1i 3993 . 2 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ (( I ↾ dom 𝐴) ∪ ( I ↾ ran 𝐴)) ⊆ 𝐵)
3 unss 4158 . 2 ((( I ↾ dom 𝐴) ⊆ 𝐵 ∧ ( I ↾ ran 𝐴) ⊆ 𝐵) ↔ (( I ↾ dom 𝐴) ∪ ( I ↾ ran 𝐴)) ⊆ 𝐵)
4 relres 5875 . . . . . 6 Rel ( I ↾ dom 𝐴)
5 ssrel 5650 . . . . . 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 3496 . . . . . . . . . 10 𝑥 ∈ V
87eldm 5762 . . . . . . . . 9 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦)
9 df-br 5058 . . . . . . . . . 10 (𝑥 I 𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ I )
10 vex 3496 . . . . . . . . . . 11 𝑧 ∈ V
1110ideq 5716 . . . . . . . . . 10 (𝑥 I 𝑧𝑥 = 𝑧)
129, 11bitr3i 279 . . . . . . . . 9 (⟨𝑥, 𝑧⟩ ∈ I ↔ 𝑥 = 𝑧)
138, 12anbi12ci 629 . . . . . . . 8 ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ I ) ↔ (𝑥 = 𝑧 ∧ ∃𝑦 𝑥𝐴𝑦))
1410opelresi 5854 . . . . . . . 8 (⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ I ))
15 19.42v 1947 . . . . . . . 8 (∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) ↔ (𝑥 = 𝑧 ∧ ∃𝑦 𝑥𝐴𝑦))
1613, 14, 153bitr4i 305 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ ( I ↾ dom 𝐴) ↔ ∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦))
17 df-br 5058 . . . . . . . 8 (𝑥𝐵𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐵)
1817bicomi 226 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ 𝐵𝑥𝐵𝑧)
1916, 18imbi12i 353 . . . . . 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 1936 . . . . . . . 8 (∀𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ (∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
2221bicomi 226 . . . . . . 7 ((∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
23222albii 1814 . . . . . 6 (∀𝑥𝑧(∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑥𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
24 alcom 2155 . . . . . . . 8 (∀𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧))
25 ancomst 467 . . . . . . . . . . . 12 (((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ((𝑥𝐴𝑦𝑥 = 𝑧) → 𝑥𝐵𝑧))
26 impexp 453 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝑥 = 𝑧) → 𝑥𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)))
2725, 26bitri 277 . . . . . . . . . . 11 (((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)))
2827albii 1813 . . . . . . . . . 10 (∀𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑧(𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)))
29 19.21v 1933 . . . . . . . . . 10 (∀𝑧(𝑥𝐴𝑦 → (𝑥 = 𝑧𝑥𝐵𝑧)) ↔ (𝑥𝐴𝑦 → ∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧)))
30 equcom 2018 . . . . . . . . . . . . . 14 (𝑥 = 𝑧𝑧 = 𝑥)
3130imbi1i 352 . . . . . . . . . . . . 13 ((𝑥 = 𝑧𝑥𝐵𝑧) ↔ (𝑧 = 𝑥𝑥𝐵𝑧))
3231albii 1813 . . . . . . . . . . . 12 (∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧) ↔ ∀𝑧(𝑧 = 𝑥𝑥𝐵𝑧))
33 breq2 5061 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (𝑥𝐵𝑧𝑥𝐵𝑥))
3433equsalvw 2003 . . . . . . . . . . . 12 (∀𝑧(𝑧 = 𝑥𝑥𝐵𝑧) ↔ 𝑥𝐵𝑥)
3532, 34bitri 277 . . . . . . . . . . 11 (∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧) ↔ 𝑥𝐵𝑥)
3635imbi2i 338 . . . . . . . . . 10 ((𝑥𝐴𝑦 → ∀𝑧(𝑥 = 𝑧𝑥𝐵𝑧)) ↔ (𝑥𝐴𝑦𝑥𝐵𝑥))
3728, 29, 363bitri 299 . . . . . . . . 9 (∀𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ (𝑥𝐴𝑦𝑥𝐵𝑥))
3837albii 1813 . . . . . . . 8 (∀𝑦𝑧((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
3924, 38bitri 277 . . . . . . 7 (∀𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
4039albii 1813 . . . . . 6 (∀𝑥𝑧𝑦((𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
4123, 40bitri 277 . . . . 5 (∀𝑥𝑧(∃𝑦(𝑥 = 𝑧𝑥𝐴𝑦) → 𝑥𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
426, 20, 413bitri 299 . . . 4 (( I ↾ dom 𝐴) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥))
43 relres 5875 . . . . . 6 Rel ( I ↾ ran 𝐴)
44 ssrel 5650 . . . . . 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 3496 . . . . . . . . . 10 𝑦 ∈ V
4746elrn 5815 . . . . . . . . 9 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦)
48 df-br 5058 . . . . . . . . . 10 (𝑦 I 𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ I )
4910ideq 5716 . . . . . . . . . 10 (𝑦 I 𝑧𝑦 = 𝑧)
5048, 49bitr3i 279 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ ∈ I ↔ 𝑦 = 𝑧)
5147, 50anbi12ci 629 . . . . . . . 8 ((𝑦 ∈ ran 𝐴 ∧ ⟨𝑦, 𝑧⟩ ∈ I ) ↔ (𝑦 = 𝑧 ∧ ∃𝑥 𝑥𝐴𝑦))
5210opelresi 5854 . . . . . . . 8 (⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) ↔ (𝑦 ∈ ran 𝐴 ∧ ⟨𝑦, 𝑧⟩ ∈ I ))
53 19.42v 1947 . . . . . . . 8 (∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) ↔ (𝑦 = 𝑧 ∧ ∃𝑥 𝑥𝐴𝑦))
5451, 52, 533bitr4i 305 . . . . . . 7 (⟨𝑦, 𝑧⟩ ∈ ( I ↾ ran 𝐴) ↔ ∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦))
55 df-br 5058 . . . . . . . 8 (𝑦𝐵𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐵)
5655bicomi 226 . . . . . . 7 (⟨𝑦, 𝑧⟩ ∈ 𝐵𝑦𝐵𝑧)
5754, 56imbi12i 353 . . . . . 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 1936 . . . . . . . 8 (∀𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ (∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
6059bicomi 226 . . . . . . 7 ((∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
61602albii 1814 . . . . . 6 (∀𝑦𝑧(∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑦𝑧𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
62 alrot3 2156 . . . . . 6 (∀𝑥𝑦𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑦𝑧𝑥((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧))
63 ancomst 467 . . . . . . . . . 10 (((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ((𝑥𝐴𝑦𝑦 = 𝑧) → 𝑦𝐵𝑧))
64 impexp 453 . . . . . . . . . 10 (((𝑥𝐴𝑦𝑦 = 𝑧) → 𝑦𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)))
6563, 64bitri 277 . . . . . . . . 9 (((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ (𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)))
6665albii 1813 . . . . . . . 8 (∀𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑧(𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)))
67 19.21v 1933 . . . . . . . 8 (∀𝑧(𝑥𝐴𝑦 → (𝑦 = 𝑧𝑦𝐵𝑧)) ↔ (𝑥𝐴𝑦 → ∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧)))
68 equcom 2018 . . . . . . . . . . . 12 (𝑦 = 𝑧𝑧 = 𝑦)
6968imbi1i 352 . . . . . . . . . . 11 ((𝑦 = 𝑧𝑦𝐵𝑧) ↔ (𝑧 = 𝑦𝑦𝐵𝑧))
7069albii 1813 . . . . . . . . . 10 (∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧) ↔ ∀𝑧(𝑧 = 𝑦𝑦𝐵𝑧))
71 breq2 5061 . . . . . . . . . . 11 (𝑧 = 𝑦 → (𝑦𝐵𝑧𝑦𝐵𝑦))
7271equsalvw 2003 . . . . . . . . . 10 (∀𝑧(𝑧 = 𝑦𝑦𝐵𝑧) ↔ 𝑦𝐵𝑦)
7370, 72bitri 277 . . . . . . . . 9 (∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧) ↔ 𝑦𝐵𝑦)
7473imbi2i 338 . . . . . . . 8 ((𝑥𝐴𝑦 → ∀𝑧(𝑦 = 𝑧𝑦𝐵𝑧)) ↔ (𝑥𝐴𝑦𝑦𝐵𝑦))
7566, 67, 743bitri 299 . . . . . . 7 (∀𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ (𝑥𝐴𝑦𝑦𝐵𝑦))
76752albii 1814 . . . . . 6 (∀𝑥𝑦𝑧((𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦))
7761, 62, 763bitr2i 301 . . . . 5 (∀𝑦𝑧(∃𝑥(𝑦 = 𝑧𝑥𝐴𝑦) → 𝑦𝐵𝑧) ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦))
7845, 58, 773bitri 299 . . . 4 (( I ↾ ran 𝐴) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦))
7942, 78anbi12i 628 . . 3 ((( I ↾ dom 𝐴) ⊆ 𝐵 ∧ ( I ↾ ran 𝐴) ⊆ 𝐵) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥) ∧ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦)))
80 19.26-2 1865 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝑥𝐵𝑥) ∧ (𝑥𝐴𝑦𝑦𝐵𝑦)) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑥) ∧ ∀𝑥𝑦(𝑥𝐴𝑦𝑦𝐵𝑦)))
81 pm4.76 521 . . . 4 (((𝑥𝐴𝑦𝑥𝐵𝑥) ∧ (𝑥𝐴𝑦𝑦𝐵𝑦)) ↔ (𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
82812albii 1814 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦𝑥𝐵𝑥) ∧ (𝑥𝐴𝑦𝑦𝐵𝑦)) ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
8379, 80, 823bitr2i 301 . 2 ((( I ↾ dom 𝐴) ⊆ 𝐵 ∧ ( I ↾ ran 𝐴) ⊆ 𝐵) ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
842, 3, 833bitr2i 301 1 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1528  wex 1773  wcel 2107  cun 3932  wss 3934  cop 4565   class class class wbr 5057   I cid 5452  dom cdm 5548  ran crn 5549  cres 5550  Rel wrel 5553
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560
This theorem is referenced by:  reflexg  39955
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