![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 1stf1 | 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) |
1stfval.p | ⊢ 𝑃 = (𝐶 1stF 𝐷) |
1stf1.p | ⊢ (𝜑 → 𝑅 ∈ 𝐵) |
Ref | Expression |
---|---|
1stf1 | ⊢ (𝜑 → ((1st ‘𝑃)‘𝑅) = (1st ‘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 | 1stfval.p | . . . . 5 ⊢ 𝑃 = (𝐶 1stF 𝐷) | |
7 | 1, 2, 3, 4, 5, 6 | 1stfval 17433 | . . . 4 ⊢ (𝜑 → 𝑃 = 〈(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))〉) |
8 | fo1st 7691 | . . . . . . 7 ⊢ 1st :V–onto→V | |
9 | fofun 6566 | . . . . . . 7 ⊢ (1st :V–onto→V → Fun 1st ) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ Fun 1st |
11 | 2 | fvexi 6659 | . . . . . 6 ⊢ 𝐵 ∈ V |
12 | resfunexg 6955 | . . . . . 6 ⊢ ((Fun 1st ∧ 𝐵 ∈ V) → (1st ↾ 𝐵) ∈ V) | |
13 | 10, 11, 12 | mp2an 691 | . . . . 5 ⊢ (1st ↾ 𝐵) ∈ V |
14 | 11, 11 | mpoex 7760 | . . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))) ∈ V |
15 | 13, 14 | op1std 7681 | . . . 4 ⊢ (𝑃 = 〈(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))〉 → (1st ‘𝑃) = (1st ↾ 𝐵)) |
16 | 7, 15 | syl 17 | . . 3 ⊢ (𝜑 → (1st ‘𝑃) = (1st ↾ 𝐵)) |
17 | 16 | fveq1d 6647 | . 2 ⊢ (𝜑 → ((1st ‘𝑃)‘𝑅) = ((1st ↾ 𝐵)‘𝑅)) |
18 | 1stf1.p | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐵) | |
19 | 18 | fvresd 6665 | . 2 ⊢ (𝜑 → ((1st ↾ 𝐵)‘𝑅) = (1st ‘𝑅)) |
20 | 17, 19 | eqtrd 2833 | 1 ⊢ (𝜑 → ((1st ‘𝑃)‘𝑅) = (1st ‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 ↾ cres 5521 Fun wfun 6318 –onto→wfo 6322 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1st c1st 7669 Basecbs 16475 Hom chom 16568 Catccat 16927 ×c cxpc 17410 1stF c1stf 17411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-ndx 16478 df-slot 16479 df-base 16481 df-hom 16581 df-cco 16582 df-xpc 17414 df-1stf 17415 |
This theorem is referenced by: prf1st 17446 1st2ndprf 17448 uncf1 17478 uncf2 17479 diag11 17485 yonedalem21 17515 yonedalem22 17520 |
Copyright terms: Public domain | W3C validator |