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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 16657 | . . . 4 ⊢ (𝜑 → 𝑄 = 〈(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))〉) |
8 | fo2nd 7080 | . . . . . . 7 ⊢ 2nd :V–onto→V | |
9 | fofun 6029 | . . . . . . 7 ⊢ (2nd :V–onto→V → Fun 2nd ) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ Fun 2nd |
11 | fvex 6113 | . . . . . . 7 ⊢ (Base‘𝑇) ∈ V | |
12 | 2, 11 | eqeltri 2684 | . . . . . 6 ⊢ 𝐵 ∈ V |
13 | resfunexg 6384 | . . . . . 6 ⊢ ((Fun 2nd ∧ 𝐵 ∈ V) → (2nd ↾ 𝐵) ∈ V) | |
14 | 10, 12, 13 | mp2an 704 | . . . . 5 ⊢ (2nd ↾ 𝐵) ∈ V |
15 | 12, 12 | mpt2ex 7136 | . . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V |
16 | 14, 15 | op1std 7069 | . . . 4 ⊢ (𝑄 = 〈(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))〉 → (1st ‘𝑄) = (2nd ↾ 𝐵)) |
17 | 7, 16 | syl 17 | . . 3 ⊢ (𝜑 → (1st ‘𝑄) = (2nd ↾ 𝐵)) |
18 | 17 | fveq1d 6105 | . 2 ⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = ((2nd ↾ 𝐵)‘𝑅)) |
19 | 2ndf1.p | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐵) | |
20 | fvres 6117 | . . 3 ⊢ (𝑅 ∈ 𝐵 → ((2nd ↾ 𝐵)‘𝑅) = (2nd ‘𝑅)) | |
21 | 19, 20 | syl 17 | . 2 ⊢ (𝜑 → ((2nd ↾ 𝐵)‘𝑅) = (2nd ‘𝑅)) |
22 | 18, 21 | eqtrd 2644 | 1 ⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = (2nd ‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ↾ cres 5040 Fun wfun 5798 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 2nd c2nd 7058 Basecbs 15695 Hom chom 15779 Catccat 16148 ×c cxpc 16631 2ndF c2ndf 16633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-hom 15793 df-cco 15794 df-xpc 16635 df-2ndf 16637 |
This theorem is referenced by: prf2nd 16668 1st2ndprf 16669 uncf1 16699 uncf2 16700 curf2ndf 16710 yonedalem21 16736 yonedalem22 16741 |
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