Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dmrelrnrel Structured version   Visualization version   GIF version

Theorem dmrelrnrel 39936
Description: A relation preserving function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
dmrelrnrel.x 𝑥𝜑
dmrelrnrel.y 𝑦𝜑
dmrelrnrel.i (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
dmrelrnrel.b (𝜑𝐵𝐴)
dmrelrnrel.c (𝜑𝐶𝐴)
dmrelrnrel.r (𝜑𝐵𝑅𝐶)
Assertion
Ref Expression
dmrelrnrel (𝜑 → (𝐹𝐵)𝑆(𝐹𝐶))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑥)

Proof of Theorem dmrelrnrel
StepHypRef Expression
1 id 22 . . 3 (𝜑𝜑)
2 dmrelrnrel.b . . 3 (𝜑𝐵𝐴)
3 dmrelrnrel.c . . 3 (𝜑𝐶𝐴)
41, 2, 3jca31 558 . 2 (𝜑 → ((𝜑𝐵𝐴) ∧ 𝐶𝐴))
5 dmrelrnrel.r . 2 (𝜑𝐵𝑅𝐶)
6 nfv 1992 . . . . 5 𝑦 𝐵𝐴
7 dmrelrnrel.y . . . . . . . 8 𝑦𝜑
87, 6nfan 1977 . . . . . . 7 𝑦(𝜑𝐵𝐴)
9 nfv 1992 . . . . . . 7 𝑦 𝐶𝐴
108, 9nfan 1977 . . . . . 6 𝑦((𝜑𝐵𝐴) ∧ 𝐶𝐴)
11 nfv 1992 . . . . . 6 𝑦(𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶))
1210, 11nfim 1974 . . . . 5 𝑦(((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶)))
136, 12nfim 1974 . . . 4 𝑦(𝐵𝐴 → (((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶))))
14 eleq1 2827 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝐴𝐶𝐴))
1514anbi2d 742 . . . . . 6 (𝑦 = 𝐶 → (((𝜑𝐵𝐴) ∧ 𝑦𝐴) ↔ ((𝜑𝐵𝐴) ∧ 𝐶𝐴)))
16 breq2 4808 . . . . . . 7 (𝑦 = 𝐶 → (𝐵𝑅𝑦𝐵𝑅𝐶))
17 fveq2 6353 . . . . . . . 8 (𝑦 = 𝐶 → (𝐹𝑦) = (𝐹𝐶))
1817breq2d 4816 . . . . . . 7 (𝑦 = 𝐶 → ((𝐹𝐵)𝑆(𝐹𝑦) ↔ (𝐹𝐵)𝑆(𝐹𝐶)))
1916, 18imbi12d 333 . . . . . 6 (𝑦 = 𝐶 → ((𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦)) ↔ (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶))))
2015, 19imbi12d 333 . . . . 5 (𝑦 = 𝐶 → ((((𝜑𝐵𝐴) ∧ 𝑦𝐴) → (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦))) ↔ (((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶)))))
2120imbi2d 329 . . . 4 (𝑦 = 𝐶 → ((𝐵𝐴 → (((𝜑𝐵𝐴) ∧ 𝑦𝐴) → (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦)))) ↔ (𝐵𝐴 → (((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶))))))
22 dmrelrnrel.x . . . . . . . 8 𝑥𝜑
23 nfv 1992 . . . . . . . 8 𝑥 𝐵𝐴
2422, 23nfan 1977 . . . . . . 7 𝑥(𝜑𝐵𝐴)
25 nfv 1992 . . . . . . 7 𝑥 𝑦𝐴
2624, 25nfan 1977 . . . . . 6 𝑥((𝜑𝐵𝐴) ∧ 𝑦𝐴)
27 nfv 1992 . . . . . 6 𝑥(𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦))
2826, 27nfim 1974 . . . . 5 𝑥(((𝜑𝐵𝐴) ∧ 𝑦𝐴) → (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦)))
29 eleq1 2827 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
3029anbi2d 742 . . . . . . 7 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜑𝐵𝐴)))
3130anbi1d 743 . . . . . 6 (𝑥 = 𝐵 → (((𝜑𝑥𝐴) ∧ 𝑦𝐴) ↔ ((𝜑𝐵𝐴) ∧ 𝑦𝐴)))
32 breq1 4807 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝑅𝑦𝐵𝑅𝑦))
33 fveq2 6353 . . . . . . . 8 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
3433breq1d 4814 . . . . . . 7 (𝑥 = 𝐵 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝐵)𝑆(𝐹𝑦)))
3532, 34imbi12d 333 . . . . . 6 (𝑥 = 𝐵 → ((𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)) ↔ (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦))))
3631, 35imbi12d 333 . . . . 5 (𝑥 = 𝐵 → ((((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ↔ (((𝜑𝐵𝐴) ∧ 𝑦𝐴) → (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦)))))
37 dmrelrnrel.i . . . . . . 7 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
3837r19.21bi 3070 . . . . . 6 ((𝜑𝑥𝐴) → ∀𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
3938r19.21bi 3070 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
4028, 36, 39vtoclg1f 3405 . . . 4 (𝐵𝐴 → (((𝜑𝐵𝐴) ∧ 𝑦𝐴) → (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦))))
4113, 21, 40vtoclg1f 3405 . . 3 (𝐶𝐴 → (𝐵𝐴 → (((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶)))))
423, 2, 41sylc 65 . 2 (𝜑 → (((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶))))
434, 5, 42mp2d 49 1 (𝜑 → (𝐹𝐵)𝑆(𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wnf 1857  wcel 2139  wral 3050   class class class wbr 4804  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057
This theorem is referenced by:  pimincfltioc  41450  pimincfltioo  41452
  Copyright terms: Public domain W3C validator