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Theorem idfu2nd 17868
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 7427 . 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 17867 . . . . 5 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
76fveq2d 6904 . . . 4 (𝜑 → (2nd𝐼) = (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩))
83fvexi 6914 . . . . . 6 𝐵 ∈ V
9 resiexg 7924 . . . . . 6 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
108, 9ax-mp 5 . . . . 5 ( I ↾ 𝐵) ∈ V
118, 8xpex 7759 . . . . . 6 (𝐵 × 𝐵) ∈ V
1211mptex 7239 . . . . 5 (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))) ∈ V
1310, 12op2nd 8006 . . . 4 (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))
147, 13eqtrdi 2783 . . 3 (𝜑 → (2nd𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))))
15 simpr 483 . . . . . 6 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
1615fveq2d 6904 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
17 df-ov 7427 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1816, 17eqtr4di 2785 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝑋𝐻𝑌))
1918reseq2d 5987 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ( I ↾ (𝐻𝑧)) = ( I ↾ (𝑋𝐻𝑌)))
20 idfu2nd.x . . . 4 (𝜑𝑋𝐵)
21 idfu2nd.y . . . 4 (𝜑𝑌𝐵)
2220, 21opelxpd 5719 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
23 ovex 7457 . . . 4 (𝑋𝐻𝑌) ∈ V
24 resiexg 7924 . . . 4 ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
2523, 24mp1i 13 . . 3 (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
2614, 19, 22, 25fvmptd 7015 . 2 (𝜑 → ((2nd𝐼)‘⟨𝑋, 𝑌⟩) = ( I ↾ (𝑋𝐻𝑌)))
271, 26eqtrid 2779 1 (𝜑 → (𝑋(2nd𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3471  cop 4636  cmpt 5233   I cid 5577   × cxp 5678  cres 5682  cfv 6551  (class class class)co 7424  2nd c2nd 7996  Basecbs 17185  Hom chom 17249  Catccat 17649  idfunccidfu 17846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-2nd 7998  df-idfu 17850
This theorem is referenced by:  idfu2  17869  idfucl  17872  cofulid  17881  cofurid  17882  idffth  17927  ressffth  17932  catciso  18105
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