MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  diag11 Structured version   Visualization version   GIF version

Theorem diag11 18235
Description: Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
diagval.l 𝐿 = (𝐶Δfunc𝐷)
diagval.c (𝜑𝐶 ∈ Cat)
diagval.d (𝜑𝐷 ∈ Cat)
diag11.a 𝐴 = (Base‘𝐶)
diag11.c (𝜑𝑋𝐴)
diag11.k 𝐾 = ((1st𝐿)‘𝑋)
diag11.b 𝐵 = (Base‘𝐷)
diag11.y (𝜑𝑌𝐵)
Assertion
Ref Expression
diag11 (𝜑 → ((1st𝐾)‘𝑌) = 𝑋)

Proof of Theorem diag11
StepHypRef Expression
1 diag11.k . . . . 5 𝐾 = ((1st𝐿)‘𝑋)
2 diagval.l . . . . . . . 8 𝐿 = (𝐶Δfunc𝐷)
3 diagval.c . . . . . . . 8 (𝜑𝐶 ∈ Cat)
4 diagval.d . . . . . . . 8 (𝜑𝐷 ∈ Cat)
52, 3, 4diagval 18232 . . . . . . 7 (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
65fveq2d 6901 . . . . . 6 (𝜑 → (1st𝐿) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
76fveq1d 6899 . . . . 5 (𝜑 → ((1st𝐿)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
81, 7eqtrid 2780 . . . 4 (𝜑𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))
98fveq2d 6901 . . 3 (𝜑 → (1st𝐾) = (1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)))
109fveq1d 6899 . 2 (𝜑 → ((1st𝐾)‘𝑌) = ((1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌))
11 eqid 2728 . . 3 (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
12 diag11.a . . 3 𝐴 = (Base‘𝐶)
13 eqid 2728 . . . 4 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
14 eqid 2728 . . . 4 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
1513, 3, 4, 141stfcl 18188 . . 3 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
16 diag11.b . . 3 𝐵 = (Base‘𝐷)
17 diag11.c . . 3 (𝜑𝑋𝐴)
18 eqid 2728 . . 3 ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋)
19 diag11.y . . 3 (𝜑𝑌𝐵)
2011, 12, 3, 4, 15, 16, 17, 18, 19curf11 18218 . 2 (𝜑 → ((1st ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))‘𝑋))‘𝑌) = (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌))
21 df-ov 7423 . . . 4 (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩)
2213, 12, 16xpcbas 18169 . . . . 5 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
23 eqid 2728 . . . . 5 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
2417, 19opelxpd 5717 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
2513, 22, 23, 3, 4, 14, 241stf1 18183 . . . 4 (𝜑 → ((1st ‘(𝐶 1stF 𝐷))‘⟨𝑋, 𝑌⟩) = (1st ‘⟨𝑋, 𝑌⟩))
2621, 25eqtrid 2780 . . 3 (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = (1st ‘⟨𝑋, 𝑌⟩))
27 op1stg 8005 . . . 4 ((𝑋𝐴𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2817, 19, 27syl2anc 583 . . 3 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2926, 28eqtrd 2768 . 2 (𝜑 → (𝑋(1st ‘(𝐶 1stF 𝐷))𝑌) = 𝑋)
3010, 20, 293eqtrd 2772 1 (𝜑 → ((1st𝐾)‘𝑌) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cop 4635   × cxp 5676  cfv 6548  (class class class)co 7420  1st c1st 7991  Basecbs 17180  Hom chom 17244  Catccat 17644   ×c cxpc 18159   1stF c1stf 18160   curryF ccurf 18202  Δfunccdiag 18204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12504  df-z 12590  df-dec 12709  df-uz 12854  df-fz 13518  df-struct 17116  df-slot 17151  df-ndx 17163  df-base 17181  df-hom 17257  df-cco 17258  df-cat 17648  df-cid 17649  df-func 17844  df-xpc 18163  df-1stf 18164  df-curf 18206  df-diag 18208
This theorem is referenced by:  curf2ndf  18239
  Copyright terms: Public domain W3C validator