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Theorem cotr2g 14331
 Description: Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 14332 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 5942 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
2 nfv 1915 . . . . . 6 𝑦 𝑥𝐷
3 nfv 1915 . . . . . 6 𝑧 𝑥𝐷
42, 319.21-2 2208 . . . . 5 (∀𝑦𝑧(𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))) ↔ (𝑥𝐷 → ∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
54albii 1821 . . . 4 (∀𝑥𝑦𝑧(𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))) ↔ ∀𝑥(𝑥𝐷 → ∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
6 simpl 486 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐵𝑦)
7 id 22 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝑦𝐴𝑧) → (𝑥𝐵𝑦𝑦𝐴𝑧))
8 simpr 488 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑦𝐴𝑧)
96, 7, 83jca 1125 . . . . . . . . . 10 ((𝑥𝐵𝑦𝑦𝐴𝑧) → (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧))
10 simp2 1134 . . . . . . . . . 10 ((𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧) → (𝑥𝐵𝑦𝑦𝐴𝑧))
119, 10impbii 212 . . . . . . . . 9 ((𝑥𝐵𝑦𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧))
12 cotr2g.d . . . . . . . . . . . 12 dom 𝐵𝐷
13 vex 3447 . . . . . . . . . . . . 13 𝑥 ∈ V
14 vex 3447 . . . . . . . . . . . . 13 𝑦 ∈ V
1513, 14breldm 5745 . . . . . . . . . . . 12 (𝑥𝐵𝑦𝑥 ∈ dom 𝐵)
1612, 15sseldi 3916 . . . . . . . . . . 11 (𝑥𝐵𝑦𝑥𝐷)
1716pm4.71ri 564 . . . . . . . . . 10 (𝑥𝐵𝑦 ↔ (𝑥𝐷𝑥𝐵𝑦))
18 cotr2g.e . . . . . . . . . . . 12 (ran 𝐵 ∩ dom 𝐴) ⊆ 𝐸
1913, 14brelrn 5780 . . . . . . . . . . . . 13 (𝑥𝐵𝑦𝑦 ∈ ran 𝐵)
20 vex 3447 . . . . . . . . . . . . . 14 𝑧 ∈ V
2114, 20breldm 5745 . . . . . . . . . . . . 13 (𝑦𝐴𝑧𝑦 ∈ dom 𝐴)
22 elin 3900 . . . . . . . . . . . . . 14 (𝑦 ∈ (ran 𝐵 ∩ dom 𝐴) ↔ (𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴))
2322biimpri 231 . . . . . . . . . . . . 13 ((𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴) → 𝑦 ∈ (ran 𝐵 ∩ dom 𝐴))
2419, 21, 23syl2an 598 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑦 ∈ (ran 𝐵 ∩ dom 𝐴))
2518, 24sseldi 3916 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑦𝐸)
2625pm4.71ri 564 . . . . . . . . . 10 ((𝑥𝐵𝑦𝑦𝐴𝑧) ↔ (𝑦𝐸 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)))
27 cotr2g.f . . . . . . . . . . . 12 ran 𝐴𝐹
2814, 20brelrn 5780 . . . . . . . . . . . 12 (𝑦𝐴𝑧𝑧 ∈ ran 𝐴)
2927, 28sseldi 3916 . . . . . . . . . . 11 (𝑦𝐴𝑧𝑧𝐹)
3029pm4.71ri 564 . . . . . . . . . 10 (𝑦𝐴𝑧 ↔ (𝑧𝐹𝑦𝐴𝑧))
3117, 26, 303anbi123i 1152 . . . . . . . . 9 ((𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧) ↔ ((𝑥𝐷𝑥𝐵𝑦) ∧ (𝑦𝐸 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)) ∧ (𝑧𝐹𝑦𝐴𝑧)))
32 3an6 1443 . . . . . . . . . 10 (((𝑥𝐷𝑥𝐵𝑦) ∧ (𝑦𝐸 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)) ∧ (𝑧𝐹𝑦𝐴𝑧)) ↔ ((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧)))
3310, 9impbii 212 . . . . . . . . . . 11 ((𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑦𝑦𝐴𝑧))
3433anbi2i 625 . . . . . . . . . 10 (((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧) ∧ 𝑦𝐴𝑧)) ↔ ((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)))
3532, 34bitri 278 . . . . . . . . 9 (((𝑥𝐷𝑥𝐵𝑦) ∧ (𝑦𝐸 ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)) ∧ (𝑧𝐹𝑦𝐴𝑧)) ↔ ((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)))
3611, 31, 353bitri 300 . . . . . . . 8 ((𝑥𝐵𝑦𝑦𝐴𝑧) ↔ ((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)))
3736imbi1i 353 . . . . . . 7 (((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)) → 𝑥𝐶𝑧))
38 impexp 454 . . . . . . 7 ((((𝑥𝐷𝑦𝐸𝑧𝐹) ∧ (𝑥𝐵𝑦𝑦𝐴𝑧)) → 𝑥𝐶𝑧) ↔ ((𝑥𝐷𝑦𝐸𝑧𝐹) → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
39 3impexp 1355 . . . . . . 7 (((𝑥𝐷𝑦𝐸𝑧𝐹) → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ (𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
4037, 38, 393bitri 300 . . . . . 6 (((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
4140albii 1821 . . . . 5 (∀𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑧(𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
42412albii 1822 . . . 4 (∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑥𝑦𝑧(𝑥𝐷 → (𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
43 df-ral 3114 . . . 4 (∀𝑥𝐷𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑥(𝑥𝐷 → ∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))))
445, 42, 433bitr4i 306 . . 3 (∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑥𝐷𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
45 df-ral 3114 . . . . . 6 (∀𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑦(𝑦𝐸 → ∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
46 19.21v 1940 . . . . . . . 8 (∀𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ (𝑦𝐸 → ∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
4746bicomi 227 . . . . . . 7 ((𝑦𝐸 → ∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
4847albii 1821 . . . . . 6 (∀𝑦(𝑦𝐸 → ∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
4945, 48bitri 278 . . . . 5 (∀𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))))
5049bicomi 227 . . . 4 (∀𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
5150ralbii 3136 . . 3 (∀𝑥𝐷𝑦𝑧(𝑦𝐸 → (𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))) ↔ ∀𝑥𝐷𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
5244, 51bitri 278 . 2 (∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑥𝐷𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
53 df-ral 3114 . . . . 5 (∀𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)))
5453bicomi 227 . . . 4 (∀𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
5554ralbii 3136 . . 3 (∀𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑦𝐸𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
5655ralbii 3136 . 2 (∀𝑥𝐷𝑦𝐸𝑧(𝑧𝐹 → ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧)) ↔ ∀𝑥𝐷𝑦𝐸𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
571, 52, 563bitri 300 1 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝐷𝑦𝐸𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   ∈ wcel 2112  ∀wral 3109   ∩ cin 3883   ⊆ wss 3884   class class class wbr 5033  dom cdm 5523  ran crn 5524   ∘ ccom 5527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534 This theorem is referenced by:  cotr2  14332
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