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| 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 17962 | . . . 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 769 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
| 12 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
| 13 | 11, 12 | oveq12d 7449 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌)) |
| 14 | 11, 12 | oveq12d 7449 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) |
| 15 | 11 | fveq2d 6910 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹‘𝑥) = (𝐹‘𝑋)) |
| 16 | 12 | fveq2d 6910 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑌)) |
| 17 | 15, 16 | oveq12d 7449 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
| 18 | 13, 14, 17 | f1eq123d 6840 | . . . 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 3614 | . . 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 3612 | . 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 1540 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 –1-1→wf1 6558 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 Func cfunc 17899 Faith cfth 17950 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-ixp 8938 df-func 17903 df-fth 17952 |
| This theorem is referenced by: fthi 17965 ffthf1o 17966 fthoppc 17970 cofth 17982 |
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