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Mirrors > Home > MPE Home > Th. List > isfth2 | Structured version Visualization version GIF version |
Description: Equivalent condition for a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
isfth.b | ⊢ 𝐵 = (Base‘𝐶) |
isfth.h | ⊢ 𝐻 = (Hom ‘𝐶) |
isfth.j | ⊢ 𝐽 = (Hom ‘𝐷) |
Ref | Expression |
---|---|
isfth2 | ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfth.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | 1 | isfth 17968 | . 2 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦))) |
3 | isfth.h | . . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | isfth.j | . . . . . . 7 ⊢ 𝐽 = (Hom ‘𝐷) | |
5 | simpll 767 | . . . . . . 7 ⊢ (((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝐹(𝐶 Func 𝐷)𝐺) | |
6 | simplr 769 | . . . . . . 7 ⊢ (((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
7 | simpr 484 | . . . . . . 7 ⊢ (((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) | |
8 | 1, 3, 4, 5, 6, 7 | funcf2 17919 | . . . . . 6 ⊢ (((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))) |
9 | df-f1 6568 | . . . . . . 7 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ Fun ◡(𝑥𝐺𝑦))) | |
10 | 9 | baib 535 | . . . . . 6 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ Fun ◡(𝑥𝐺𝑦))) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ Fun ◡(𝑥𝐺𝑦))) |
12 | 11 | ralbidva 3174 | . . . 4 ⊢ ((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦))) |
13 | 12 | ralbidva 3174 | . . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦))) |
14 | 13 | pm5.32i 574 | . 2 ⊢ ((𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦))) ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦))) |
15 | 2, 14 | bitr4i 278 | 1 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ◡ccnv 5688 Fun wfun 6557 ⟶wf 6559 –1-1→wf1 6560 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Hom chom 17309 Func cfunc 17905 Faith cfth 17957 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-ixp 8937 df-func 17909 df-fth 17959 |
This theorem is referenced by: isffth2 17970 fthf1 17971 cofth 17989 fthestrcsetc 18206 fthsetcestrc 18221 thincfth 48848 |
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