Theorem dmrelrnrel 44602

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		id 22	. . 3 ⊢ (𝜑 → 𝜑)
2		dmrelrnrel.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
3		dmrelrnrel.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
4	1, 2, 3	jca31 513	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴))
5		dmrelrnrel.r	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
6		nfv 1909	. . . . 5 ⊢ Ⅎ𝑦 𝐵 ∈ 𝐴
7		dmrelrnrel.y	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
8	7, 6	nfan 1894	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐵 ∈ 𝐴)
9		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑦 𝐶 ∈ 𝐴
10	8, 9	nfan 1894	. . . . . 6 ⊢ Ⅎ𝑦((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴)
11		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑦(𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))
12	10, 11	nfim 1891	. . . . 5 ⊢ Ⅎ𝑦(((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶)))
13	6, 12	nfim 1891	. . . 4 ⊢ Ⅎ𝑦(𝐵 ∈ 𝐴 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))))
14		eleq1 2816	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
15	14	anbi2d 628	. . . . . 6 ⊢ (𝑦 = 𝐶 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ↔ ((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴)))
16		breq2 5154	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶))
17		fveq2 6900	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝐹‘𝑦) = (𝐹‘𝐶))
18	17	breq2d 5162	. . . . . . 7 ⊢ (𝑦 = 𝐶 → ((𝐹‘𝐵)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝐵)𝑆(𝐹‘𝐶)))
19	16, 18	imbi12d 343	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦)) ↔ (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))))
20	15, 19	imbi12d 343	. . . . 5 ⊢ (𝑦 = 𝐶 → ((((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦))) ↔ (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶)))))
21	20	imbi2d 339	. . . 4 ⊢ (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))))))
22		dmrelrnrel.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
23		nfv 1909	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐵 ∈ 𝐴
24	22, 23	nfan 1894	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐵 ∈ 𝐴)
25		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
26	24, 25	nfan 1894	. . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴)
27		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑥(𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦))
28	26, 27	nfim 1891	. . . . 5 ⊢ Ⅎ𝑥(((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦)))
29		eleq1 2816	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
30	29	anbi2d 628	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝐵 ∈ 𝐴)))
31	30	anbi1d 629	. . . . . 6 ⊢ (𝑥 = 𝐵 → (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ↔ ((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴)))
32		breq1 5153	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦))
33		fveq2 6900	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
34	33	breq1d 5160	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝐵)𝑆(𝐹‘𝑦)))
35	32, 34	imbi12d 343	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)) ↔ (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦))))
36	31, 35	imbi12d 343	. . . . 5 ⊢ (𝑥 = 𝐵 → ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ↔ (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦)))))
37		dmrelrnrel.i	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
38	37	r19.21bi 3244	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
39	38	r19.21bi 3244	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
40	28, 36, 39	vtoclg1f 3556	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦))))
41	13, 21, 40	vtoclg1f 3556	. . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶)))))
42	3, 2, 41	sylc 65	. 2 ⊢ (𝜑 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))))
43	4, 5, 42	mp2d 49	1 ⊢ (𝜑 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  ∀wral 3057   class class class wbr 5150  ‘cfv 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-iota 6503  df-fv 6559

This theorem is referenced by:  pimincfltioc  46106  pimincfltioo  46108