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Theorem cotr2g 15025
Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 15026 for the main application. (Contributed by RP, 22-Mar-2020.)
Hypotheses
Ref Expression
cotr2g.d dom 𝐵 ⊆ 𝐷
cotr2g.e (ran 𝐵 ∩ dom 𝐴) ⊆ 𝐸
cotr2g.f ran 𝐴 ⊆ 𝐹
Assertion
Ref Expression
cotr2g ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧   𝑦,𝐸,𝑧   𝑧,𝐹
Allowed substitution hints:   𝐸(𝑥)   𝐹(𝑥,𝑦)
Proof of Theorem cotr2g
StepHypRef Expression
1 cotrg 6139 . 2 ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
2 nfv 1913 . . . . . 6 Ⅎ𝑦 𝑥 ∈ 𝐷
3 nfv 1913 . . . . . 6 Ⅎ𝑧 𝑥 ∈ 𝐷
42, 319.21-2 2210 . . . . 5 (∀𝑦∀𝑧(𝑥 ∈ 𝐷 → (𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))) ↔ (𝑥 ∈ 𝐷 → ∀𝑦∀𝑧(𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
54albii 1817 . . . 4 (∀𝑥∀𝑦∀𝑧(𝑥 ∈ 𝐷 → (𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))) ↔ ∀𝑥(𝑥 ∈ 𝐷 → ∀𝑦∀𝑧(𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
6 simpl 482 . . . . . . . . . . 11 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐵𝑦)
7 id 22 . . . . . . . . . . 11 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
8 simpr 484 . . . . . . . . . . 11 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑦𝐴𝑧)
96, 7, 83jca 1128 . . . . . . . . . 10 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧))
10 simp2 1137 . . . . . . . . . 10 ((𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧) → (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
119, 10impbii 209 . . . . . . . . 9 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧))
12 cotr2g.d . . . . . . . . . . . 12 dom 𝐵 ⊆ 𝐷
13 vex 3492 . . . . . . . . . . . . 13 𝑥 ∈ V
14 vex 3492 . . . . . . . . . . . . 13 𝑦 ∈ V
1513, 14breldm 5933 . . . . . . . . . . . 12 (𝑥𝐵𝑦 → 𝑥 ∈ dom 𝐵)
1612, 15sselid 4006 . . . . . . . . . . 11 (𝑥𝐵𝑦 → 𝑥 ∈ 𝐷)
1716pm4.71ri 560 . . . . . . . . . 10 (𝑥𝐵𝑦 ↔ (𝑥 ∈ 𝐷 ∧ 𝑥𝐵𝑦))
18 cotr2g.e . . . . . . . . . . . 12 (ran 𝐵 ∩ dom 𝐴) ⊆ 𝐸
1913, 14brelrn 5967 . . . . . . . . . . . . 13 (𝑥𝐵𝑦 → 𝑦 ∈ ran 𝐵)
20 vex 3492 . . . . . . . . . . . . . 14 𝑧 ∈ V
2114, 20breldm 5933 . . . . . . . . . . . . 13 (𝑦𝐴𝑧 → 𝑦 ∈ dom 𝐴)
22 elin 3992 . . . . . . . . . . . . . 14 (𝑦 ∈ (ran 𝐵 ∩ dom 𝐴) ↔ (𝑦 ∈ ran 𝐵 ∧ 𝑦 ∈ dom 𝐴))
2322biimpri 228 . . . . . . . . . . . . 13 ((𝑦 ∈ ran 𝐵 ∧ 𝑦 ∈ dom 𝐴) → 𝑦 ∈ (ran 𝐵 ∩ dom 𝐴))
2419, 21, 23syl2an 595 . . . . . . . . . . . 12 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑦 ∈ (ran 𝐵 ∩ dom 𝐴))
2518, 24sselid 4006 . . . . . . . . . . 11 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑦 ∈ 𝐸)
2625pm4.71ri 560 . . . . . . . . . 10 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ (𝑦 ∈ 𝐸 ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
27 cotr2g.f . . . . . . . . . . . 12 ran 𝐴 ⊆ 𝐹
2814, 20brelrn 5967 . . . . . . . . . . . 12 (𝑦𝐴𝑧 → 𝑧 ∈ ran 𝐴)
2927, 28sselid 4006 . . . . . . . . . . 11 (𝑦𝐴𝑧 → 𝑧 ∈ 𝐹)
3029pm4.71ri 560 . . . . . . . . . 10 (𝑦𝐴𝑧 ↔ (𝑧 ∈ 𝐹 ∧ 𝑦𝐴𝑧))
3117, 26, 303anbi123i 1155 . . . . . . . . 9 ((𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑥𝐵𝑦) ∧ (𝑦 ∈ 𝐸 ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑦𝐴𝑧)))
32 3an6 1446 . . . . . . . . . 10 (((𝑥 ∈ 𝐷 ∧ 𝑥𝐵𝑦) ∧ (𝑦 ∈ 𝐸 ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑦𝐴𝑧)) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸 ∧ 𝑧 ∈ 𝐹) ∧ (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧)))
3310, 9impbii 209 . . . . . . . . . . 11 ((𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
3433anbi2i 622 . . . . . . . . . 10 (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸 ∧ 𝑧 ∈ 𝐹) ∧ (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧)) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸 ∧ 𝑧 ∈ 𝐹) ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
3532, 34bitri 275 . . . . . . . . 9 (((𝑥 ∈ 𝐷 ∧ 𝑥𝐵𝑦) ∧ (𝑦 ∈ 𝐸 ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑦𝐴𝑧)) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸 ∧ 𝑧 ∈ 𝐹) ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
3611, 31, 353bitri 297 . . . . . . . 8 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸 ∧ 𝑧 ∈ 𝐹) ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
3736imbi1i 349 . . . . . . 7 (((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸 ∧ 𝑧 ∈ 𝐹) ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)) → 𝑥𝐶𝑧))
38 impexp 450 . . . . . . 7 ((((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸 ∧ 𝑧 ∈ 𝐹) ∧ (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)) → 𝑥𝐶𝑧) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸 ∧ 𝑧 ∈ 𝐹) → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
39 3impexp 1358 . . . . . . 7 (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸 ∧ 𝑧 ∈ 𝐹) → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ (𝑥 ∈ 𝐷 → (𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
4037, 38, 393bitri 297 . . . . . 6 (((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (𝑥 ∈ 𝐷 → (𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
4140albii 1817 . . . . 5 (∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑧(𝑥 ∈ 𝐷 → (𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
42412albii 1818 . . . 4 (∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑥∀𝑦∀𝑧(𝑥 ∈ 𝐷 → (𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
43 df-ral 3068 . . . 4 (∀𝑥 ∈ 𝐷 ∀𝑦∀𝑧(𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑥(𝑥 ∈ 𝐷 → ∀𝑦∀𝑧(𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
445, 42, 433bitr4i 303 . . 3 (∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑥 ∈ 𝐷 ∀𝑦∀𝑧(𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
45 df-ral 3068 . . . . . 6 (∀𝑦 ∈ 𝐸 ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑦(𝑦 ∈ 𝐸 → ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
46 19.21v 1938 . . . . . . . 8 (∀𝑧(𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ (𝑦 ∈ 𝐸 → ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
4746bicomi 224 . . . . . . 7 ((𝑦 ∈ 𝐸 → ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑧(𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
4847albii 1817 . . . . . 6 (∀𝑦(𝑦 ∈ 𝐸 → ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑦∀𝑧(𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
4945, 48bitri 275 . . . . 5 (∀𝑦 ∈ 𝐸 ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑦∀𝑧(𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
5049bicomi 224 . . . 4 (∀𝑦∀𝑧(𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑦 ∈ 𝐸 ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
5150ralbii 3099 . . 3 (∀𝑥 ∈ 𝐷 ∀𝑦∀𝑧(𝑦 ∈ 𝐸 → (𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
5244, 51bitri 275 . 2 (∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
53 df-ral 3068 . . . . 5 (∀𝑧 ∈ 𝐹 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
5453bicomi 224 . . . 4 (∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑧 ∈ 𝐹 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
5554ralbii 3099 . . 3 (∀𝑦 ∈ 𝐸 ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
5655ralbii 3099 . 2 (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧(𝑧 ∈ 𝐹 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
571, 52, 563bitri 297 1 ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1535   ∈ wcel 2108  ∀wral 3067   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166  dom cdm 5700  ran crn 5701   ∘ ccom 5704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711
This theorem is referenced by:  cotr2  15026
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