Theorem fthmon 16587

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 2622	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2622	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
4		eqid 2622	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
5		fthmon.n	. . . . . 6 ⊢ 𝑁 = (Mono‘𝐷)
6		fthmon.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
7		fthfunc 16567	. . . . . . . . . . . 12 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
8	7	ssbri 4697	. . . . . . . . . . 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 4654	. . . . . . . . . 10 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
11	9, 10	sylib 208	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12		funcrcl 16523	. . . . . . . . 9 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
13	11, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
14	13	simprd 479	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
15	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐷 ∈ Cat)
16		fthmon.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
17	9	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Func 𝐷)𝐺)
18	16, 2, 17	funcf1 16526	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹:𝐵⟶(Base‘𝐷))
19		fthmon.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
20	19	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑋 ∈ 𝐵)
21	18, 20	ffvelrnd 6360	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹‘𝑋) ∈ (Base‘𝐷))
22		fthmon.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
23	22	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑌 ∈ 𝐵)
24	18, 23	ffvelrnd 6360	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹‘𝑌) ∈ (Base‘𝐷))
25		simpr1 1067	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑧 ∈ 𝐵)
26	18, 25	ffvelrnd 6360	. . . . . 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 481	. . . . . 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 16528	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑧𝐺𝑋):(𝑧𝐻𝑋)⟶((𝐹‘𝑧)(Hom ‘𝐷)(𝐹‘𝑋)))
31		simpr2 1068	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑓 ∈ (𝑧𝐻𝑋))
32	30, 31	ffvelrnd 6360	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑓) ∈ ((𝐹‘𝑧)(Hom ‘𝐷)(𝐹‘𝑋)))
33		simpr3 1069	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
34	30, 33	ffvelrnd 6360	. . . . . 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 16396	. . . . 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 2622	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
37	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑅 ∈ (𝑋𝐻𝑌))
38	16, 29, 36, 4, 17, 25, 20, 23, 31, 37	funcco 16531	. . . . . . 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 16531	. . . . . . 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 2637	. . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Faith 𝐷)𝐺)
42	13	simpld 475	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
43	42	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
44	16, 29, 36, 43, 25, 20, 23, 31, 37	catcocl 16346	. . . . . . 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 16346	. . . . . . 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 16578	. . . . . 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 270	. . . . 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 16578	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑋)‘𝑓) = ((𝑧𝐺𝑋)‘𝑔) ↔ 𝑓 = 𝑔))
49	35, 47, 48	3bitr3d 298	. . . 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 219	. . 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 2972	. 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 16394	. 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 957	1 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝑀𝑌))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ⟨cop 4183   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Hom chom 15952  compcco 15953  Catccat 16325  Monocmon 16388   Func cfunc 16514   Faith cfth 16563

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-cat 16329  df-mon 16390  df-func 16518  df-fth 16565

This theorem is referenced by:  fthepi  16588