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Mirrors > Home > MPE Home > Th. List > fullfo | Structured version Visualization version GIF version |
Description: The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
isfull.b | ⊢ 𝐵 = (Base‘𝐶) |
isfull.j | ⊢ 𝐽 = (Hom ‘𝐷) |
isfull.h | ⊢ 𝐻 = (Hom ‘𝐶) |
fullfo.f | ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺) |
fullfo.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
fullfo.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
fullfo | ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fullfo.f | . . 3 ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺) | |
2 | isfull.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
3 | isfull.j | . . . . 5 ⊢ 𝐽 = (Hom ‘𝐷) | |
4 | isfull.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
5 | 2, 3, 4 | isfull2 17965 | . . . 4 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) |
6 | 5 | simprbi 496 | . . 3 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))) |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))) |
8 | fullfo.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | fullfo.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 6911 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹‘𝑥) = (𝐹‘𝑋)) |
16 | 12 | fveq2d 6911 | . . . . . 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 | foeq123d 6842 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))) |
19 | 10, 18 | rspcdv 3614 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))) |
20 | 8, 19 | rspcimdv 3612 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))) |
21 | 7, 20 | mpd 15 | 1 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Hom chom 17309 Func cfunc 17905 Full cful 17956 |
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-fo 6569 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-full 17958 |
This theorem is referenced by: fulli 17967 ffthf1o 17973 fulloppc 17976 cofull 17988 |
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