Theorem dmrelrnrel 38414

Description: A relation preserving function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
dmrelrnrel.x	⊢ Ⅎ𝑥𝜑
dmrelrnrel.y	⊢ Ⅎ𝑦𝜑
dmrelrnrel.i	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
dmrelrnrel.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)
dmrelrnrel.c	⊢ (𝜑 → 𝐶 ∈ 𝐴)
dmrelrnrel.r	⊢ (𝜑 → 𝐵𝑅𝐶)

Assertion

Ref	Expression
dmrelrnrel	⊢ (𝜑 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))

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

Step	Hyp	Ref	Expression
1		dmrelrnrel.r	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
2		id 22	. . . 4 ⊢ (𝜑 → 𝜑)
3		dmrelrnrel.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
4		dmrelrnrel.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
5	2, 3, 4	jca31 555	. . 3 ⊢ (𝜑 → ((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴))
6		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑦𝐶
7		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑦 𝐵 ∈ 𝐴
8		dmrelrnrel.y	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
9	8, 7	nfan 1816	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐵 ∈ 𝐴)
10		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑦 𝐶 ∈ 𝐴
11	9, 10	nfan 1816	. . . . . . 7 ⊢ Ⅎ𝑦((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴)
12		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑦(𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))
13	11, 12	nfim 1813	. . . . . 6 ⊢ Ⅎ𝑦(((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶)))
14	7, 13	nfim 1813	. . . . 5 ⊢ Ⅎ𝑦(𝐵 ∈ 𝐴 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))))
15		eleq1 2676	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
16	15	anbi2d 736	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ↔ ((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴)))
17		breq2 4587	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶))
18		fveq2 6103	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → (𝐹‘𝑦) = (𝐹‘𝐶))
19	18	breq2d 4595	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → ((𝐹‘𝐵)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝐵)𝑆(𝐹‘𝐶)))
20	17, 19	imbi12d 333	. . . . . . 7 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦)) ↔ (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))))
21	16, 20	imbi12d 333	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦))) ↔ (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶)))))
22	21	imbi2d 329	. . . . 5 ⊢ (𝑦 = 𝐶 → ((𝐵 ∈ 𝐴 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦)))) ↔ (𝐵 ∈ 𝐴 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))))))
23		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥𝐵
24		dmrelrnrel.x	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜑
25		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐵 ∈ 𝐴
26	24, 25	nfan 1816	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐵 ∈ 𝐴)
27		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
28	26, 27	nfan 1816	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴)
29		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑥(𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦))
30	28, 29	nfim 1813	. . . . . 6 ⊢ Ⅎ𝑥(((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦)))
31		eleq1 2676	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
32	31	anbi2d 736	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝐵 ∈ 𝐴)))
33	32	anbi1d 737	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ↔ ((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴)))
34		breq1 4586	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦))
35		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
36	35	breq1d 4593	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝐵)𝑆(𝐹‘𝑦)))
37	34, 36	imbi12d 333	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)) ↔ (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦))))
38	33, 37	imbi12d 333	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ↔ (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦)))))
39		dmrelrnrel.i	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
40	39	r19.21bi 2916	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
41	40	r19.21bi 2916	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
42	23, 30, 38, 41	vtoclgf 3237	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦))))
43	6, 14, 22, 42	vtoclgf 3237	. . . 4 ⊢ (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶)))))
44	4, 3, 43	sylc 63	. . 3 ⊢ (𝜑 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))))
45	5, 44	mpd 15	. 2 ⊢ (𝜑 → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶)))
46	1, 45	mpd 15	1 ⊢ (𝜑 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))

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