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Theorem idfu2nd 16741
Description: Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
idfuval.i 𝐼 = (idfunc𝐶)
idfuval.b 𝐵 = (Base‘𝐶)
idfuval.c (𝜑𝐶 ∈ Cat)
idfuval.h 𝐻 = (Hom ‘𝐶)
idfu2nd.x (𝜑𝑋𝐵)
idfu2nd.y (𝜑𝑌𝐵)
Assertion
Ref Expression
idfu2nd (𝜑 → (𝑋(2nd𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))

Proof of Theorem idfu2nd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ov 6877 . 2 (𝑋(2nd𝐼)𝑌) = ((2nd𝐼)‘⟨𝑋, 𝑌⟩)
2 idfuval.i . . . . . 6 𝐼 = (idfunc𝐶)
3 idfuval.b . . . . . 6 𝐵 = (Base‘𝐶)
4 idfuval.c . . . . . 6 (𝜑𝐶 ∈ Cat)
5 idfuval.h . . . . . 6 𝐻 = (Hom ‘𝐶)
62, 3, 4, 5idfuval 16740 . . . . 5 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
76fveq2d 6412 . . . 4 (𝜑 → (2nd𝐼) = (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩))
83fvexi 6422 . . . . . 6 𝐵 ∈ V
9 resiexg 7332 . . . . . 6 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
108, 9ax-mp 5 . . . . 5 ( I ↾ 𝐵) ∈ V
118, 8xpex 7192 . . . . . 6 (𝐵 × 𝐵) ∈ V
1211mptex 6711 . . . . 5 (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))) ∈ V
1310, 12op2nd 7407 . . . 4 (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))
147, 13syl6eq 2856 . . 3 (𝜑 → (2nd𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))))
15 simpr 473 . . . . . 6 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
1615fveq2d 6412 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
17 df-ov 6877 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1816, 17syl6eqr 2858 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝑋𝐻𝑌))
1918reseq2d 5597 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ( I ↾ (𝐻𝑧)) = ( I ↾ (𝑋𝐻𝑌)))
20 idfu2nd.x . . . 4 (𝜑𝑋𝐵)
21 idfu2nd.y . . . 4 (𝜑𝑌𝐵)
22 opelxpi 5348 . . . 4 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
2320, 21, 22syl2anc 575 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
24 ovex 6906 . . . 4 (𝑋𝐻𝑌) ∈ V
25 resiexg 7332 . . . 4 ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
2624, 25mp1i 13 . . 3 (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
2714, 19, 23, 26fvmptd 6509 . 2 (𝜑 → ((2nd𝐼)‘⟨𝑋, 𝑌⟩) = ( I ↾ (𝑋𝐻𝑌)))
281, 27syl5eq 2852 1 (𝜑 → (𝑋(2nd𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  Vcvv 3391  cop 4376  cmpt 4923   I cid 5218   × cxp 5309  cres 5313  cfv 6101  (class class class)co 6874  2nd c2nd 7397  Basecbs 16068  Hom chom 16164  Catccat 16529  idfunccidfu 16719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-2nd 7399  df-idfu 16723
This theorem is referenced by:  idfu2  16742  idfucl  16745  cofulid  16754  cofurid  16755  idffth  16797  ressffth  16802  catciso  16961
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