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

Theorem dmrelrnrel 38414
 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 dmrelrnrel.r . 2 (𝜑𝐵𝑅𝐶)
2 id 22 . . . 4 (𝜑𝜑)
3 dmrelrnrel.b . . . 4 (𝜑𝐵𝐴)
4 dmrelrnrel.c . . . 4 (𝜑𝐶𝐴)
52, 3, 4jca31 555 . . 3 (𝜑 → ((𝜑𝐵𝐴) ∧ 𝐶𝐴))
6 nfcv 2751 . . . . 5 𝑦𝐶
7 nfv 1830 . . . . . 6 𝑦 𝐵𝐴
8 dmrelrnrel.y . . . . . . . . 9 𝑦𝜑
98, 7nfan 1816 . . . . . . . 8 𝑦(𝜑𝐵𝐴)
10 nfv 1830 . . . . . . . 8 𝑦 𝐶𝐴
119, 10nfan 1816 . . . . . . 7 𝑦((𝜑𝐵𝐴) ∧ 𝐶𝐴)
12 nfv 1830 . . . . . . 7 𝑦(𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶))
1311, 12nfim 1813 . . . . . 6 𝑦(((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶)))
147, 13nfim 1813 . . . . 5 𝑦(𝐵𝐴 → (((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶))))
15 eleq1 2676 . . . . . . . 8 (𝑦 = 𝐶 → (𝑦𝐴𝐶𝐴))
1615anbi2d 736 . . . . . . 7 (𝑦 = 𝐶 → (((𝜑𝐵𝐴) ∧ 𝑦𝐴) ↔ ((𝜑𝐵𝐴) ∧ 𝐶𝐴)))
17 breq2 4587 . . . . . . . 8 (𝑦 = 𝐶 → (𝐵𝑅𝑦𝐵𝑅𝐶))
18 fveq2 6103 . . . . . . . . 9 (𝑦 = 𝐶 → (𝐹𝑦) = (𝐹𝐶))
1918breq2d 4595 . . . . . . . 8 (𝑦 = 𝐶 → ((𝐹𝐵)𝑆(𝐹𝑦) ↔ (𝐹𝐵)𝑆(𝐹𝐶)))
2017, 19imbi12d 333 . . . . . . 7 (𝑦 = 𝐶 → ((𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦)) ↔ (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶))))
2116, 20imbi12d 333 . . . . . 6 (𝑦 = 𝐶 → ((((𝜑𝐵𝐴) ∧ 𝑦𝐴) → (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦))) ↔ (((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶)))))
2221imbi2d 329 . . . . 5 (𝑦 = 𝐶 → ((𝐵𝐴 → (((𝜑𝐵𝐴) ∧ 𝑦𝐴) → (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦)))) ↔ (𝐵𝐴 → (((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶))))))
23 nfcv 2751 . . . . . 6 𝑥𝐵
24 dmrelrnrel.x . . . . . . . . 9 𝑥𝜑
25 nfv 1830 . . . . . . . . 9 𝑥 𝐵𝐴
2624, 25nfan 1816 . . . . . . . 8 𝑥(𝜑𝐵𝐴)
27 nfv 1830 . . . . . . . 8 𝑥 𝑦𝐴
2826, 27nfan 1816 . . . . . . 7 𝑥((𝜑𝐵𝐴) ∧ 𝑦𝐴)
29 nfv 1830 . . . . . . 7 𝑥(𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦))
3028, 29nfim 1813 . . . . . 6 𝑥(((𝜑𝐵𝐴) ∧ 𝑦𝐴) → (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦)))
31 eleq1 2676 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
3231anbi2d 736 . . . . . . . 8 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜑𝐵𝐴)))
3332anbi1d 737 . . . . . . 7 (𝑥 = 𝐵 → (((𝜑𝑥𝐴) ∧ 𝑦𝐴) ↔ ((𝜑𝐵𝐴) ∧ 𝑦𝐴)))
34 breq1 4586 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝑅𝑦𝐵𝑅𝑦))
35 fveq2 6103 . . . . . . . . 9 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
3635breq1d 4593 . . . . . . . 8 (𝑥 = 𝐵 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝐵)𝑆(𝐹𝑦)))
3734, 36imbi12d 333 . . . . . . 7 (𝑥 = 𝐵 → ((𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)) ↔ (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦))))
3833, 37imbi12d 333 . . . . . 6 (𝑥 = 𝐵 → ((((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))) ↔ (((𝜑𝐵𝐴) ∧ 𝑦𝐴) → (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦)))))
39 dmrelrnrel.i . . . . . . . 8 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
4039r19.21bi 2916 . . . . . . 7 ((𝜑𝑥𝐴) → ∀𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
4140r19.21bi 2916 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
4223, 30, 38, 41vtoclgf 3237 . . . . 5 (𝐵𝐴 → (((𝜑𝐵𝐴) ∧ 𝑦𝐴) → (𝐵𝑅𝑦 → (𝐹𝐵)𝑆(𝐹𝑦))))
436, 14, 22, 42vtoclgf 3237 . . . 4 (𝐶𝐴 → (𝐵𝐴 → (((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶)))))
444, 3, 43sylc 63 . . 3 (𝜑 → (((𝜑𝐵𝐴) ∧ 𝐶𝐴) → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶))))
455, 44mpd 15 . 2 (𝜑 → (𝐵𝑅𝐶 → (𝐹𝐵)𝑆(𝐹𝐶)))
461, 45mpd 15 1 (𝜑 → (𝐹𝐵)𝑆(𝐹𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812 This theorem is referenced by:  pimincfltioc  39603  pimincfltioo  39605
 Copyright terms: Public domain W3C validator