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Mirrors > Home > MPE Home > Th. List > fthi | 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) |
fthi.r | ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌)) |
fthi.s | ⊢ (𝜑 → 𝑆 ∈ (𝑋𝐻𝑌)) |
Ref | Expression |
---|---|
fthi | ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfth.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | isfth.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
3 | isfth.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
4 | fthf1.f | . . 3 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) | |
5 | fthf1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | fthf1.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | fthf1 17181 | . 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
8 | fthi.r | . 2 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌)) | |
9 | fthi.s | . 2 ⊢ (𝜑 → 𝑆 ∈ (𝑋𝐻𝑌)) | |
10 | f1fveq 7014 | . 2 ⊢ (((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∧ (𝑅 ∈ (𝑋𝐻𝑌) ∧ 𝑆 ∈ (𝑋𝐻𝑌))) → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆)) | |
11 | 7, 8, 9, 10 | syl12anc 834 | 1 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 –1-1→wf1 6346 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Hom chom 16570 Faith cfth 17167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-ixp 8456 df-func 17122 df-fth 17169 |
This theorem is referenced by: fthsect 17189 fthmon 17191 |
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