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Mirrors > Home > MPE Home > Th. List > isfull2 | Structured version Visualization version GIF version |
Description: Equivalent condition for a full functor. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
isfull.b | ⊢ 𝐵 = (Base‘𝐶) |
isfull.j | ⊢ 𝐽 = (Hom ‘𝐷) |
isfull.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
isfull2 | ⊢ (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfull.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | isfull.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
3 | 1, 2 | isfull 17861 | . 2 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
4 | isfull.h | . . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶) | |
5 | simpll 766 | . . . . . . 7 ⊢ (((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝐹(𝐶 Func 𝐷)𝐺) | |
6 | simplr 768 | . . . . . . 7 ⊢ (((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
7 | simpr 486 | . . . . . . 7 ⊢ (((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) | |
8 | 1, 4, 2, 5, 6, 7 | funcf2 17818 | . . . . . 6 ⊢ (((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))) |
9 | ffn 6718 | . . . . . 6 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) → (𝑥𝐺𝑦) Fn (𝑥𝐻𝑦)) | |
10 | df-fo 6550 | . . . . . . 7 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) | |
11 | 10 | baib 537 | . . . . . 6 ⊢ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
12 | 8, 9, 11 | 3syl 18 | . . . . 5 ⊢ (((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
13 | 12 | ralbidva 3176 | . . . 4 ⊢ ((𝐹(𝐶 Func 𝐷)𝐺 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
14 | 13 | ralbidva 3176 | . . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
15 | 14 | pm5.32i 576 | . 2 ⊢ ((𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))) ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
16 | 3, 15 | bitr4i 278 | 1 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 class class class wbr 5149 ran crn 5678 Fn wfn 6539 ⟶wf 6540 –onto→wfo 6542 ‘cfv 6544 (class class class)co 7409 Basecbs 17144 Hom chom 17208 Func cfunc 17804 Full cful 17853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-map 8822 df-ixp 8892 df-func 17808 df-full 17855 |
This theorem is referenced by: fullfo 17863 isffth2 17867 cofull 17885 fullestrcsetc 18103 fullsetcestrc 18118 fullthinc 47666 |
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