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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 17183 | . . . 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 499 | . . 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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝐵) |
11 | simplr 767 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
12 | simpr 487 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
13 | 11, 12 | oveq12d 7176 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌)) |
14 | 11, 12 | oveq12d 7176 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌)) |
15 | 11 | fveq2d 6676 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹‘𝑥) = (𝐹‘𝑋)) |
16 | 12 | fveq2d 6676 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑌)) |
17 | 15, 16 | oveq12d 7176 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
18 | 13, 14, 17 | foeq123d 6611 | . . . 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 3617 | . . 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 3615 | . 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 class class class wbr 5068 –onto→wfo 6355 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Hom chom 16578 Func cfunc 17126 Full cful 17174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-ixp 8464 df-func 17130 df-full 17176 |
This theorem is referenced by: fulli 17185 ffthf1o 17191 fulloppc 17194 cofull 17206 |
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