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Theorem diag2 18139
Description: Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
diag2.l 𝐿 = (𝐶Δfunc𝐷)
diag2.a 𝐴 = (Base‘𝐶)
diag2.b 𝐵 = (Base‘𝐷)
diag2.h 𝐻 = (Hom ‘𝐶)
diag2.c (𝜑𝐶 ∈ Cat)
diag2.d (𝜑𝐷 ∈ Cat)
diag2.x (𝜑𝑋𝐴)
diag2.y (𝜑𝑌𝐴)
diag2.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
diag2 (𝜑 → ((𝑋(2nd𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))

Proof of Theorem diag2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 diag2.l . . . . . 6 𝐿 = (𝐶Δfunc𝐷)
2 diag2.c . . . . . 6 (𝜑𝐶 ∈ Cat)
3 diag2.d . . . . . 6 (𝜑𝐷 ∈ Cat)
41, 2, 3diagval 18134 . . . . 5 (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
54fveq2d 6847 . . . 4 (𝜑 → (2nd𝐿) = (2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))))
65oveqd 7375 . . 3 (𝜑 → (𝑋(2nd𝐿)𝑌) = (𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌))
76fveq1d 6845 . 2 (𝜑 → ((𝑋(2nd𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹))
8 eqid 2733 . . 3 (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)) = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷))
9 diag2.a . . 3 𝐴 = (Base‘𝐶)
10 eqid 2733 . . . 4 (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
11 eqid 2733 . . . 4 (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷)
1210, 2, 3, 111stfcl 18090 . . 3 (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶))
13 diag2.b . . 3 𝐵 = (Base‘𝐷)
14 diag2.h . . 3 𝐻 = (Hom ‘𝐶)
15 eqid 2733 . . 3 (Id‘𝐷) = (Id‘𝐷)
16 diag2.x . . 3 (𝜑𝑋𝐴)
17 diag2.y . . 3 (𝜑𝑌𝐴)
18 diag2.f . . 3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
19 eqid 2733 . . 3 ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹)
208, 9, 2, 3, 12, 13, 14, 15, 16, 17, 18, 19curf2 18123 . 2 (𝜑 → ((𝑋(2nd ‘(⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))𝑌)‘𝐹) = (𝑥𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))))
2110, 9, 13xpcbas 18071 . . . . . . 7 (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
22 eqid 2733 . . . . . . 7 (Hom ‘(𝐶 ×c 𝐷)) = (Hom ‘(𝐶 ×c 𝐷))
232adantr 482 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐶 ∈ Cat)
243adantr 482 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
25 opelxpi 5671 . . . . . . . 8 ((𝑋𝐴𝑥𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
2616, 25sylan 581 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝑋, 𝑥⟩ ∈ (𝐴 × 𝐵))
27 opelxpi 5671 . . . . . . . 8 ((𝑌𝐴𝑥𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
2817, 27sylan 581 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝑌, 𝑥⟩ ∈ (𝐴 × 𝐵))
2910, 21, 22, 23, 24, 11, 26, 281stf2 18086 . . . . . 6 ((𝜑𝑥𝐵) → (⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩) = (1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩)))
3029oveqd 7375 . . . . 5 ((𝜑𝑥𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)))
31 df-ov 7361 . . . . . 6 (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = ((1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩)
3218adantr 482 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝐹 ∈ (𝑋𝐻𝑌))
33 eqid 2733 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
34 simpr 486 . . . . . . . . . 10 ((𝜑𝑥𝐵) → 𝑥𝐵)
3513, 33, 15, 24, 34catidcl 17567 . . . . . . . . 9 ((𝜑𝑥𝐵) → ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥))
3632, 35opelxpd 5672 . . . . . . . 8 ((𝜑𝑥𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
3716adantr 482 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑋𝐴)
3817adantr 482 . . . . . . . . 9 ((𝜑𝑥𝐵) → 𝑌𝐴)
3910, 9, 13, 14, 33, 37, 34, 38, 34, 22xpchom2 18079 . . . . . . . 8 ((𝜑𝑥𝐵) → (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩) = ((𝑋𝐻𝑌) × (𝑥(Hom ‘𝐷)𝑥)))
4036, 39eleqtrrd 2837 . . . . . . 7 ((𝜑𝑥𝐵) → ⟨𝐹, ((Id‘𝐷)‘𝑥)⟩ ∈ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))
4140fvresd 6863 . . . . . 6 ((𝜑𝑥𝐵) → ((1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
4231, 41eqtrid 2785 . . . . 5 ((𝜑𝑥𝐵) → (𝐹(1st ↾ (⟨𝑋, 𝑥⟩(Hom ‘(𝐶 ×c 𝐷))⟨𝑌, 𝑥⟩))((Id‘𝐷)‘𝑥)) = (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩))
43 op1stg 7934 . . . . . 6 ((𝐹 ∈ (𝑋𝐻𝑌) ∧ ((Id‘𝐷)‘𝑥) ∈ (𝑥(Hom ‘𝐷)𝑥)) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
4418, 35, 43syl2an2r 684 . . . . 5 ((𝜑𝑥𝐵) → (1st ‘⟨𝐹, ((Id‘𝐷)‘𝑥)⟩) = 𝐹)
4530, 42, 443eqtrd 2777 . . . 4 ((𝜑𝑥𝐵) → (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥)) = 𝐹)
4645mpteq2dva 5206 . . 3 (𝜑 → (𝑥𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝑥𝐵𝐹))
47 fconstmpt 5695 . . 3 (𝐵 × {𝐹}) = (𝑥𝐵𝐹)
4846, 47eqtr4di 2791 . 2 (𝜑 → (𝑥𝐵 ↦ (𝐹(⟨𝑋, 𝑥⟩(2nd ‘(𝐶 1stF 𝐷))⟨𝑌, 𝑥⟩)((Id‘𝐷)‘𝑥))) = (𝐵 × {𝐹}))
497, 20, 483eqtrd 2777 1 (𝜑 → ((𝑋(2nd𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {csn 4587  cop 4593  cmpt 5189   × cxp 5632  cres 5636  cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  Basecbs 17088  Hom chom 17149  Catccat 17549  Idccid 17550   ×c cxpc 18061   1stF c1stf 18062   curryF ccurf 18104  Δfunccdiag 18106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-cat 17553  df-cid 17554  df-func 17749  df-xpc 18065  df-1stf 18066  df-curf 18108  df-diag 18110
This theorem is referenced by:  diag2cl  18140
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