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Theorem idfu2nd 17891
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 7419 . 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 17890 . . . . 5 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
76fveq2d 6897 . . . 4 (𝜑 → (2nd𝐼) = (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩))
83fvexi 6907 . . . . . 6 𝐵 ∈ V
9 resiexg 7917 . . . . . 6 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
108, 9ax-mp 5 . . . . 5 ( I ↾ 𝐵) ∈ V
118, 8xpex 7753 . . . . . 6 (𝐵 × 𝐵) ∈ V
1211mptex 7232 . . . . 5 (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))) ∈ V
1310, 12op2nd 8004 . . . 4 (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))
147, 13eqtrdi 2782 . . 3 (𝜑 → (2nd𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))))
15 simpr 483 . . . . . 6 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
1615fveq2d 6897 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
17 df-ov 7419 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1816, 17eqtr4di 2784 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝑋𝐻𝑌))
1918reseq2d 5981 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ( I ↾ (𝐻𝑧)) = ( I ↾ (𝑋𝐻𝑌)))
20 idfu2nd.x . . . 4 (𝜑𝑋𝐵)
21 idfu2nd.y . . . 4 (𝜑𝑌𝐵)
2220, 21opelxpd 5713 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
23 ovex 7449 . . . 4 (𝑋𝐻𝑌) ∈ V
24 resiexg 7917 . . . 4 ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
2523, 24mp1i 13 . . 3 (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
2614, 19, 22, 25fvmptd 7008 . 2 (𝜑 → ((2nd𝐼)‘⟨𝑋, 𝑌⟩) = ( I ↾ (𝑋𝐻𝑌)))
271, 26eqtrid 2778 1 (𝜑 → (𝑋(2nd𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  cop 4629  cmpt 5228   I cid 5571   × cxp 5672  cres 5676  cfv 6546  (class class class)co 7416  2nd c2nd 7994  Basecbs 17208  Hom chom 17272  Catccat 17672  idfunccidfu 17869
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 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-2nd 7996  df-idfu 17873
This theorem is referenced by:  idfu2  17892  idfucl  17895  cofulid  17904  cofurid  17905  idffth  17950  ressffth  17955  catciso  18128
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