Theorem dmrelrnrel 39419

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 557	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴))
5		dmrelrnrel.r	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
6		nfv 1843	. . . . 5 ⊢ Ⅎ𝑦 𝐵 ∈ 𝐴
7		dmrelrnrel.y	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
8	7, 6	nfan 1828	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐵 ∈ 𝐴)
9		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑦 𝐶 ∈ 𝐴
10	8, 9	nfan 1828	. . . . . 6 ⊢ Ⅎ𝑦((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴)
11		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑦(𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))
12	10, 11	nfim 1825	. . . . 5 ⊢ Ⅎ𝑦(((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶)))
13	6, 12	nfim 1825	. . . 4 ⊢ Ⅎ𝑦(𝐵 ∈ 𝐴 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))))
14		eleq1 2689	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
15	14	anbi2d 740	. . . . . 6 ⊢ (𝑦 = 𝐶 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ↔ ((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ 𝐴)))
16		breq2 4657	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶))
17		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝐹‘𝑦) = (𝐹‘𝐶))
18	17	breq2d 4665	. . . . . . 7 ⊢ (𝑦 = 𝐶 → ((𝐹‘𝐵)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝐵)𝑆(𝐹‘𝐶)))
19	16, 18	imbi12d 334	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦)) ↔ (𝐵𝑅𝐶 → (𝐹‘𝐵)𝑆(𝐹‘𝐶))))
20	15, 19	imbi12d 334	. . . . 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 330	. . . 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 1843	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐵 ∈ 𝐴
24	22, 23	nfan 1828	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐵 ∈ 𝐴)
25		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
26	24, 25	nfan 1828	. . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴)
27		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑥(𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦))
28	26, 27	nfim 1825	. . . . 5 ⊢ Ⅎ𝑥(((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦)))
29		eleq1 2689	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
30	29	anbi2d 740	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝐵 ∈ 𝐴)))
31	30	anbi1d 741	. . . . . 6 ⊢ (𝑥 = 𝐵 → (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) ↔ ((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴)))
32		breq1 4656	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦))
33		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
34	33	breq1d 4663	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝐵)𝑆(𝐹‘𝑦)))
35	32, 34	imbi12d 334	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)) ↔ (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦))))
36	31, 35	imbi12d 334	. . . . 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 2932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
39	38	r19.21bi 2932	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
40	28, 36, 39	vtoclg1f 3265	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (((𝜑 ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐵𝑅𝑦 → (𝐹‘𝐵)𝑆(𝐹‘𝑦))))
41	13, 21, 40	vtoclg1f 3265	. . 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 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  ∀wral 2912   class class class wbr 4653  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  pimincfltioc  40926  pimincfltioo  40928