Theorem fthmon 17879

Description: A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
fthmon.b	⊢ 𝐵 = (Base‘𝐶)
fthmon.h	⊢ 𝐻 = (Hom ‘𝐶)
fthmon.f	⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
fthmon.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
fthmon.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
fthmon.r	⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
fthmon.m	⊢ 𝑀 = (Mono‘𝐶)
fthmon.n	⊢ 𝑁 = (Mono‘𝐷)
fthmon.1	⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹‘𝑋)𝑁(𝐹‘𝑌)))

Assertion

Ref	Expression
fthmon	⊢ (𝜑 → 𝑅 ∈ (𝑋𝑀𝑌))

Proof of Theorem fthmon

Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fthmon.r	. 2 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))
2		eqid 2724	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2724	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2724	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
5		fthmon.n	. . . . . 6 ⊢ 𝑁 = (Mono‘𝐷)
6		fthmon.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
7		fthfunc 17859	. . . . . . . . . . . 12 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
8	7	ssbri 5183	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
9	6, 8	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
10		df-br 5139	. . . . . . . . . 10 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
11	9, 10	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12		funcrcl 17812	. . . . . . . . 9 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
13	11, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
14	13	simprd 495	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐷 ∈ Cat)
16		fthmon.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
17	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Func 𝐷)𝐺)
18	16, 2, 17	funcf1 17815	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹:𝐵⟶(Base‘𝐷))
19		fthmon.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
20	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑋 ∈ 𝐵)
21	18, 20	ffvelcdmd 7077	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹‘𝑋) ∈ (Base‘𝐷))
22		fthmon.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
23	22	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑌 ∈ 𝐵)
24	18, 23	ffvelcdmd 7077	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹‘𝑌) ∈ (Base‘𝐷))
25		simpr1 1191	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑧 ∈ 𝐵)
26	18, 25	ffvelcdmd 7077	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹‘𝑧) ∈ (Base‘𝐷))
27		fthmon.1	. . . . . . 7 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹‘𝑋)𝑁(𝐹‘𝑌)))
28	27	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹‘𝑋)𝑁(𝐹‘𝑌)))
29		fthmon.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶)
30	16, 29, 3, 17, 25, 20	funcf2 17817	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑧𝐺𝑋):(𝑧𝐻𝑋)⟶((𝐹‘𝑧)(Hom ‘𝐷)(𝐹‘𝑋)))
31		simpr2 1192	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑓 ∈ (𝑧𝐻𝑋))
32	30, 31	ffvelcdmd 7077	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑓) ∈ ((𝐹‘𝑧)(Hom ‘𝐷)(𝐹‘𝑋)))
33		simpr3 1193	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
34	30, 33	ffvelcdmd 7077	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑔) ∈ ((𝐹‘𝑧)(Hom ‘𝐷)(𝐹‘𝑋)))
35	2, 3, 4, 5, 15, 21, 24, 26, 28, 32, 34	moni 17682	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹‘𝑧), (𝐹‘𝑋)⟩(comp‘𝐷)(𝐹‘𝑌))((𝑧𝐺𝑋)‘𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹‘𝑧), (𝐹‘𝑋)⟩(comp‘𝐷)(𝐹‘𝑌))((𝑧𝐺𝑋)‘𝑔)) ↔ ((𝑧𝐺𝑋)‘𝑓) = ((𝑧𝐺𝑋)‘𝑔)))
36		eqid 2724	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
37	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑅 ∈ (𝑋𝐻𝑌))
38	16, 29, 36, 4, 17, 25, 20, 23, 31, 37	funcco 17820	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹‘𝑧), (𝐹‘𝑋)⟩(comp‘𝐷)(𝐹‘𝑌))((𝑧𝐺𝑋)‘𝑓)))
39	16, 29, 36, 4, 17, 25, 20, 23, 33, 37	funcco 17820	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹‘𝑧), (𝐹‘𝑋)⟩(comp‘𝐷)(𝐹‘𝑌))((𝑧𝐺𝑋)‘𝑔)))
40	38, 39	eqeq12d 2740	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓)) = ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) ↔ (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹‘𝑧), (𝐹‘𝑋)⟩(comp‘𝐷)(𝐹‘𝑌))((𝑧𝐺𝑋)‘𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹‘𝑧), (𝐹‘𝑋)⟩(comp‘𝐷)(𝐹‘𝑌))((𝑧𝐺𝑋)‘𝑔))))
41	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Faith 𝐷)𝐺)
42	13	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
43	42	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
44	16, 29, 36, 43, 25, 20, 23, 31, 37	catcocl 17628	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) ∈ (𝑧𝐻𝑌))
45	16, 29, 36, 43, 25, 20, 23, 33, 37	catcocl 17628	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) ∈ (𝑧𝐻𝑌))
46	16, 29, 3, 41, 25, 23, 44, 45	fthi 17870	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓)) = ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) ↔ (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)))
47	40, 46	bitr3d 281	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹‘𝑧), (𝐹‘𝑋)⟩(comp‘𝐷)(𝐹‘𝑌))((𝑧𝐺𝑋)‘𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹‘𝑧), (𝐹‘𝑋)⟩(comp‘𝐷)(𝐹‘𝑌))((𝑧𝐺𝑋)‘𝑔)) ↔ (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)))
48	16, 29, 3, 41, 25, 20, 31, 33	fthi 17870	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑋)‘𝑓) = ((𝑧𝐺𝑋)‘𝑔) ↔ 𝑓 = 𝑔))
49	35, 47, 48	3bitr3d 309	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) ↔ 𝑓 = 𝑔))
50	49	biimpd 228	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) → 𝑓 = 𝑔))
51	50	ralrimivvva 3195	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑧𝐻𝑋)∀𝑔 ∈ (𝑧𝐻𝑋)((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) → 𝑓 = 𝑔))
52		fthmon.m	. . 3 ⊢ 𝑀 = (Mono‘𝐶)
53	16, 29, 36, 52, 42, 19, 22	ismon2 17680	. 2 ⊢ (𝜑 → (𝑅 ∈ (𝑋𝑀𝑌) ↔ (𝑅 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑧𝐻𝑋)∀𝑔 ∈ (𝑧𝐻𝑋)((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) → 𝑓 = 𝑔))))
54	1, 51, 53	mpbir2and 710	1 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝑀𝑌))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3053  ⟨cop 4626   class class class wbr 5138  ‘cfv 6533  (class class class)co 7401  Basecbs 17143  Hom chom 17207  compcco 17208  Catccat 17607  Monocmon 17674   Func cfunc 17803   Faith cfth 17855

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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8818  df-ixp 8888  df-cat 17611  df-mon 17676  df-func 17807  df-fth 17857

This theorem is referenced by:  fthepi  17880