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Theorem cotr2g 13925
Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 13926 for the main application. (Contributed by RP, 22-Mar-2020.)
Hypotheses
Ref Expression
cotr2g.d dom 𝐵𝐷
cotr2g.e (ran 𝐵 ∩ dom 𝐴) ⊆ 𝐸
cotr2g.f ran 𝐴𝐹
Assertion
Ref Expression
cotr2g ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝐷𝑦𝐸𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 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 5648 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
2 nfv 1995 . . . . . 6 𝑦 𝑥𝐷
3 nfv 1995 . . . . . 6 𝑧 𝑥𝐷
42, 319.21-2 2234 . . . . 5 (∀𝑦𝑧(𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))) ↔ (𝑥𝐷 → ∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
54albii 1895 . . . 4 (∀𝑥𝑦𝑧(𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))) ↔ ∀𝑥(𝑥𝐷 → ∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
6 simpl 468 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐵𝑦)
7 id 22 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝑦𝐴𝑧) → (𝑥𝐵𝑦𝑦𝐴𝑧))
8 simpr 471 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑦𝐴𝑧)
96, 7, 83jca 1122 . . . . . . . . . 10 ((𝑥𝐵𝑦𝑦𝐴𝑧) → (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧))
10 simp2 1131 . . . . . . . . . 10 ((𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧) → (𝑥𝐵𝑦𝑦𝐴𝑧))
119, 10impbii 199 . . . . . . . . 9 ((𝑥𝐵𝑦𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧))
12 cotr2g.d . . . . . . . . . . . 12 dom 𝐵𝐷
13 vex 3354 . . . . . . . . . . . . 13 𝑥 ∈ V
14 vex 3354 . . . . . . . . . . . . 13 𝑦 ∈ V
1513, 14breldm 5467 . . . . . . . . . . . 12 (𝑥𝐵𝑦𝑥 ∈ dom 𝐵)
1612, 15sseldi 3750 . . . . . . . . . . 11 (𝑥𝐵𝑦𝑥𝐷)
1716pm4.71ri 550 . . . . . . . . . 10 (𝑥𝐵𝑦 ↔ (𝑥𝐷𝑥𝐵𝑦))
18 cotr2g.e . . . . . . . . . . . 12 (ran 𝐵 ∩ dom 𝐴) ⊆ 𝐸
1913, 14brelrn 5494 . . . . . . . . . . . . 13 (𝑥𝐵𝑦𝑦 ∈ ran 𝐵)
20 vex 3354 . . . . . . . . . . . . . 14 𝑧 ∈ V
2114, 20breldm 5467 . . . . . . . . . . . . 13 (𝑦𝐴𝑧𝑦 ∈ dom 𝐴)
22 elin 3947 . . . . . . . . . . . . . 14 (𝑦 ∈ (ran 𝐵 ∩ dom 𝐴) ↔ (𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴))
2322biimpri 218 . . . . . . . . . . . . 13 ((𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴) → 𝑦 ∈ (ran 𝐵 ∩ dom 𝐴))
2419, 21, 23syl2an 583 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑦 ∈ (ran 𝐵 ∩ dom 𝐴))
2518, 24sseldi 3750 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑦𝐸)
2625pm4.71ri 550 . . . . . . . . . 10 ((𝑥𝐵𝑦𝑦𝐴𝑧) ↔ (𝑦𝐸 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)))
27 cotr2g.f . . . . . . . . . . . 12 ran 𝐴𝐹
2814, 20brelrn 5494 . . . . . . . . . . . 12 (𝑦𝐴𝑧𝑧 ∈ ran 𝐴)
2927, 28sseldi 3750 . . . . . . . . . . 11 (𝑦𝐴𝑧𝑧𝐹)
3029pm4.71ri 550 . . . . . . . . . 10 (𝑦𝐴𝑧 ↔ (𝑧𝐹𝑦𝐴𝑧))
3117, 26, 303anbi123i 1158 . . . . . . . . 9 ((𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧) ↔ ((𝑥𝐷𝑥𝐵𝑦) ∧ (𝑦𝐸 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)) ∧ (𝑧𝐹𝑦𝐴𝑧)))
32 3an6 1557 . . . . . . . . . 10 (((𝑥𝐷𝑥𝐵𝑦) ∧ (𝑦𝐸 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)) ∧ (𝑧𝐹𝑦𝐴𝑧)) ↔ ((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧)))
3310, 9impbii 199 . . . . . . . . . . 11 ((𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦𝑦𝐴𝑧))
3433anbi2i 609 . . . . . . . . . 10 (((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧)) ↔ ((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)))
3532, 34bitri 264 . . . . . . . . 9 (((𝑥𝐷𝑥𝐵𝑦) ∧ (𝑦𝐸 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)) ∧ (𝑧𝐹𝑦𝐴𝑧)) ↔ ((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)))
3611, 31, 353bitri 286 . . . . . . . 8 ((𝑥𝐵𝑦𝑦𝐴𝑧) ↔ ((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)))
3736imbi1i 338 . . . . . . 7 (((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)) → 𝑥𝐶𝑧))
38 impexp 437 . . . . . . 7 ((((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)) → 𝑥𝐶𝑧) ↔ ((𝑥𝐷𝑦𝐸𝑧𝐹) → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
39 3impexp 1451 . . . . . . 7 (((𝑥𝐷𝑦𝐸𝑧𝐹) → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ (𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
4037, 38, 393bitri 286 . . . . . 6 (((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
4140albii 1895 . . . . 5 (∀𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑧(𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
42412albii 1896 . . . 4 (∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑥𝑦𝑧(𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
43 df-ral 3066 . . . 4 (∀𝑥𝐷𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑥(𝑥𝐷 → ∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
445, 42, 433bitr4i 292 . . 3 (∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑥𝐷𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
45 df-ral 3066 . . . . . 6 (∀𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑦(𝑦𝐸 → ∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
46 19.21v 2020 . . . . . . . 8 (∀𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ (𝑦𝐸 → ∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
4746bicomi 214 . . . . . . 7 ((𝑦𝐸 → ∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
4847albii 1895 . . . . . 6 (∀𝑦(𝑦𝐸 → ∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
4945, 48bitri 264 . . . . 5 (∀𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
5049bicomi 214 . . . 4 (∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
5150ralbii 3129 . . 3 (∀𝑥𝐷𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑥𝐷𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
5244, 51bitri 264 . 2 (∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑥𝐷𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
53 df-ral 3066 . . . . 5 (∀𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
5453bicomi 214 . . . 4 (∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
5554ralbii 3129 . . 3 (∀𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑦𝐸𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
5655ralbii 3129 . 2 (∀𝑥𝐷𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑥𝐷𝑦𝐸𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
571, 52, 563bitri 286 1 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝐷𝑦𝐸𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  wal 1629  wcel 2145  wral 3061  cin 3722  wss 3723   class class class wbr 4786  dom cdm 5249  ran crn 5250  ccom 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260
This theorem is referenced by:  cotr2  13926
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