![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fthf1 | Structured version Visualization version GIF version |
Description: The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
isfth.b | ⊢ 𝐵 = (Base‘𝐶) |
isfth.h | ⊢ 𝐻 = (Hom ‘𝐶) |
isfth.j | ⊢ 𝐽 = (Hom ‘𝐷) |
fthf1.f | ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) |
fthf1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
fthf1.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
fthf1 | ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fthf1.f | . . 3 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) | |
2 | isfth.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
3 | isfth.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | isfth.j | . . . . 5 ⊢ 𝐽 = (Hom ‘𝐷) | |
5 | 2, 3, 4 | isfth2 17873 | . . . 4 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
6 | 5 | simprbi 496 | . . 3 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦))) |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦))) |
8 | fthf1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | fthf1.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵) |
11 | simplr 766 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
12 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
13 | 11, 12 | oveq12d 7420 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌)) |
14 | 11, 12 | oveq12d 7420 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) |
15 | 11 | fveq2d 6886 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹‘𝑥) = (𝐹‘𝑋)) |
16 | 12 | fveq2d 6886 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑌)) |
17 | 15, 16 | oveq12d 7420 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
18 | 13, 14, 17 | f1eq123d 6816 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))) |
19 | 10, 18 | rspcdv 3596 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))) |
20 | 8, 19 | rspcimdv 3594 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))) |
21 | 7, 20 | mpd 15 | 1 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 class class class wbr 5139 –1-1→wf1 6531 ‘cfv 6534 (class class class)co 7402 Basecbs 17149 Hom chom 17213 Func cfunc 17809 Faith cfth 17861 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
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-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-map 8819 df-ixp 8889 df-func 17813 df-fth 17863 |
This theorem is referenced by: fthi 17876 ffthf1o 17877 fthoppc 17881 cofth 17893 |
Copyright terms: Public domain | W3C validator |