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Mirrors > Home > MPE Home > Th. List > diag11 | Structured version Visualization version GIF version |
Description: Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.) |
Ref | Expression |
---|---|
diagval.l | ⊢ 𝐿 = (𝐶Δfunc𝐷) |
diagval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
diagval.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
diag11.a | ⊢ 𝐴 = (Base‘𝐶) |
diag11.c | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
diag11.k | ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) |
diag11.b | ⊢ 𝐵 = (Base‘𝐷) |
diag11.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
diag11 | ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diag11.k | . . . . 5 ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) | |
2 | diagval.l | . . . . . . . 8 ⊢ 𝐿 = (𝐶Δfunc𝐷) | |
3 | diagval.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | diagval.d | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
5 | 2, 3, 4 | diagval 16703 | . . . . . . 7 ⊢ (𝜑 → 𝐿 = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) |
6 | 5 | fveq2d 6107 | . . . . . 6 ⊢ (𝜑 → (1st ‘𝐿) = (1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))) |
7 | 6 | fveq1d 6105 | . . . . 5 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋)) |
8 | 1, 7 | syl5eq 2656 | . . . 4 ⊢ (𝜑 → 𝐾 = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋)) |
9 | 8 | fveq2d 6107 | . . 3 ⊢ (𝜑 → (1st ‘𝐾) = (1st ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))) |
10 | 9 | fveq1d 6105 | . 2 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = ((1st ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌)) |
11 | eqid 2610 | . . 3 ⊢ (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)) | |
12 | diag11.a | . . 3 ⊢ 𝐴 = (Base‘𝐶) | |
13 | eqid 2610 | . . . 4 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
14 | eqid 2610 | . . . 4 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
15 | 13, 3, 4, 14 | 1stfcl 16660 | . . 3 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶)) |
16 | diag11.b | . . 3 ⊢ 𝐵 = (Base‘𝐷) | |
17 | diag11.c | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
18 | eqid 2610 | . . 3 ⊢ ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋) = ((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋) | |
19 | diag11.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
20 | 11, 12, 3, 4, 15, 16, 17, 18, 19 | curf11 16689 | . 2 ⊢ (𝜑 → ((1st ‘((1st ‘(〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌) = (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌)) |
21 | df-ov 6552 | . . . 4 ⊢ (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = ((1st ‘(𝐶 1stF 𝐷))‘〈𝑋, 𝑌〉) | |
22 | 13, 12, 16 | xpcbas 16641 | . . . . 5 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷)) |
23 | eqid 2610 | . . . . 5 ⊢ (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷)) | |
24 | opelxpi 5072 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) | |
25 | 17, 19, 24 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐴 × 𝐵)) |
26 | 13, 22, 23, 3, 4, 14, 25 | 1stf1 16655 | . . . 4 ⊢ (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘〈𝑋, 𝑌〉) = (1st ‘〈𝑋, 𝑌〉)) |
27 | 21, 26 | syl5eq 2656 | . . 3 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = (1st ‘〈𝑋, 𝑌〉)) |
28 | op1stg 7071 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) | |
29 | 17, 19, 28 | syl2anc 691 | . . 3 ⊢ (𝜑 → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
30 | 27, 29 | eqtrd 2644 | . 2 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = 𝑋) |
31 | 10, 20, 30 | 3eqtrd 2648 | 1 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 Basecbs 15695 Hom chom 15779 Catccat 16148 ×c cxpc 16631 1stF c1stf 16632 curryF ccurf 16673 Δfunccdiag 16675 |
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-rmo 2904 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-map 7746 df-ixp 7795 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-cat 16152 df-cid 16153 df-func 16341 df-xpc 16635 df-1stf 16636 df-curf 16677 df-diag 16679 |
This theorem is referenced by: curf2ndf 16710 |
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