Theorem fthmon 17972

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 2763	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2763	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2763	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
5		fthmon.n	. . . . . 6 ⊢ 𝑁 = (Mono‘𝐷)
6		fthmon.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
7		fthfunc 17952	. . . . . . . . . . . 12 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
8	7	ssbri 5146	. . . . . . . . . . 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 5102	. . . . . . . . . 10 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
11	9, 10	sylib 220	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12		funcrcl 17906	. . . . . . . . 9 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
13	11, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
14	13	simprd 499	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
15	14	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐷 ∈ Cat)
16		fthmon.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
17	9	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Func 𝐷)𝐺)
18	16, 2, 17	funcf1 17909	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹:𝐵⟶(Base‘𝐷))
19		fthmon.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
20	19	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑋 ∈ 𝐵)
21	18, 20	ffvelcdmd 7066	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹‘𝑋) ∈ (Base‘𝐷))
22		fthmon.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
23	22	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑌 ∈ 𝐵)
24	18, 23	ffvelcdmd 7066	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹‘𝑌) ∈ (Base‘𝐷))
25		simpr1 1209	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑧 ∈ 𝐵)
26	18, 25	ffvelcdmd 7066	. . . . . 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 484	. . . . . 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 17911	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑧𝐺𝑋):(𝑧𝐻𝑋)⟶((𝐹‘𝑧)(Hom ‘𝐷)(𝐹‘𝑋)))
31		simpr2 1210	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑓 ∈ (𝑧𝐻𝑋))
32	30, 31	ffvelcdmd 7066	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑓) ∈ ((𝐹‘𝑧)(Hom ‘𝐷)(𝐹‘𝑋)))
33		simpr3 1211	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
34	30, 33	ffvelcdmd 7066	. . . . . 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 17779	. . . . 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 2763	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
37	1	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑅 ∈ (𝑋𝐻𝑌))
38	16, 29, 36, 4, 17, 25, 20, 23, 31, 37	funcco 17914	. . . . . . 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 17914	. . . . . . 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 2779	. . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Faith 𝐷)𝐺)
42	13	simpld 498	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
43	42	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
44	16, 29, 36, 43, 25, 20, 23, 31, 37	catcocl 17727	. . . . . . 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 17727	. . . . . . 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 17963	. . . . . 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 283	. . . . 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 17963	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑋)‘𝑓) = ((𝑧𝐺𝑋)‘𝑔) ↔ 𝑓 = 𝑔))
49	35, 47, 48	3bitr3d 311	. . . 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 231	. . 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 3209	. 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 17777	. 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 723	1 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝑀𝑌))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∀wral 3077  ⟨cop 4589   class class class wbr 5101  ‘cfv 6521  (class class class)co 7396  Basecbs 17255  Hom chom 17307  compcco 17308  Catccat 17706  Monocmon 17771   Func cfunc 17897   Faith cfth 17948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810  df-ixp 8880  df-cat 17710  df-mon 17773  df-func 17901  df-fth 17950

This theorem is referenced by:  fthepi  17973