![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 2ndf1 | Structured version Visualization version GIF version |
Description: Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
1stfval.t | ⊢ 𝑇 = (𝐶 ×c 𝐷) |
1stfval.b | ⊢ 𝐵 = (Base‘𝑇) |
1stfval.h | ⊢ 𝐻 = (Hom ‘𝑇) |
1stfval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
1stfval.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
2ndfval.p | ⊢ 𝑄 = (𝐶 2ndF 𝐷) |
2ndf1.p | ⊢ (𝜑 → 𝑅 ∈ 𝐵) |
Ref | Expression |
---|---|
2ndf1 | ⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = (2nd ‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1stfval.t | . . . . 5 ⊢ 𝑇 = (𝐶 ×c 𝐷) | |
2 | 1stfval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑇) | |
3 | 1stfval.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝑇) | |
4 | 1stfval.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | 1stfval.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
6 | 2ndfval.p | . . . . 5 ⊢ 𝑄 = (𝐶 2ndF 𝐷) | |
7 | 1, 2, 3, 4, 5, 6 | 2ndfval 18154 | . . . 4 ⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩) |
8 | fo2nd 7990 | . . . . . . 7 ⊢ 2nd :V–onto→V | |
9 | fofun 6797 | . . . . . . 7 ⊢ (2nd :V–onto→V → Fun 2nd ) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ Fun 2nd |
11 | 2 | fvexi 6896 | . . . . . 6 ⊢ 𝐵 ∈ V |
12 | resfunexg 7209 | . . . . . 6 ⊢ ((Fun 2nd ∧ 𝐵 ∈ V) → (2nd ↾ 𝐵) ∈ V) | |
13 | 10, 11, 12 | mp2an 689 | . . . . 5 ⊢ (2nd ↾ 𝐵) ∈ V |
14 | 11, 11 | mpoex 8060 | . . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V |
15 | 13, 14 | op1std 7979 | . . . 4 ⊢ (𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ → (1st ‘𝑄) = (2nd ↾ 𝐵)) |
16 | 7, 15 | syl 17 | . . 3 ⊢ (𝜑 → (1st ‘𝑄) = (2nd ↾ 𝐵)) |
17 | 16 | fveq1d 6884 | . 2 ⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = ((2nd ↾ 𝐵)‘𝑅)) |
18 | 2ndf1.p | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐵) | |
19 | 18 | fvresd 6902 | . 2 ⊢ (𝜑 → ((2nd ↾ 𝐵)‘𝑅) = (2nd ‘𝑅)) |
20 | 17, 19 | eqtrd 2764 | 1 ⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = (2nd ‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ⟨cop 4627 ↾ cres 5669 Fun wfun 6528 –onto→wfo 6532 ‘cfv 6534 (class class class)co 7402 ∈ cmpo 7404 1st c1st 7967 2nd c2nd 7968 Basecbs 17149 Hom chom 17213 Catccat 17613 ×c cxpc 18128 2ndF c2ndf 18130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-slot 17120 df-ndx 17132 df-base 17150 df-hom 17226 df-cco 17227 df-xpc 18132 df-2ndf 18134 |
This theorem is referenced by: prf2nd 18165 1st2ndprf 18166 uncf1 18197 uncf2 18198 curf2ndf 18208 yonedalem21 18234 yonedalem22 18239 |
Copyright terms: Public domain | W3C validator |