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Theorem idfu2nd 17771
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 7364 . 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 17770 . . . . 5 (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)
76fveq2d 6850 . . . 4 (𝜑 → (2nd𝐼) = (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩))
83fvexi 6860 . . . . . 6 𝐵 ∈ V
9 resiexg 7855 . . . . . 6 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
108, 9ax-mp 5 . . . . 5 ( I ↾ 𝐵) ∈ V
118, 8xpex 7691 . . . . . 6 (𝐵 × 𝐵) ∈ V
1211mptex 7177 . . . . 5 (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))) ∈ V
1310, 12op2nd 7934 . . . 4 (2nd ‘⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))
147, 13eqtrdi 2789 . . 3 (𝜑 → (2nd𝐼) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧))))
15 simpr 486 . . . . . 6 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
1615fveq2d 6850 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
17 df-ov 7364 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1816, 17eqtr4di 2791 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝑋𝐻𝑌))
1918reseq2d 5941 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ( I ↾ (𝐻𝑧)) = ( I ↾ (𝑋𝐻𝑌)))
20 idfu2nd.x . . . 4 (𝜑𝑋𝐵)
21 idfu2nd.y . . . 4 (𝜑𝑌𝐵)
2220, 21opelxpd 5675 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
23 ovex 7394 . . . 4 (𝑋𝐻𝑌) ∈ V
24 resiexg 7855 . . . 4 ((𝑋𝐻𝑌) ∈ V → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
2523, 24mp1i 13 . . 3 (𝜑 → ( I ↾ (𝑋𝐻𝑌)) ∈ V)
2614, 19, 22, 25fvmptd 6959 . 2 (𝜑 → ((2nd𝐼)‘⟨𝑋, 𝑌⟩) = ( I ↾ (𝑋𝐻𝑌)))
271, 26eqtrid 2785 1 (𝜑 → (𝑋(2nd𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3447  cop 4596  cmpt 5192   I cid 5534   × cxp 5635  cres 5639  cfv 6500  (class class class)co 7361  2nd c2nd 7924  Basecbs 17091  Hom chom 17152  Catccat 17552  idfunccidfu 17749
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-2nd 7926  df-idfu 17753
This theorem is referenced by:  idfu2  17772  idfucl  17775  cofulid  17784  cofurid  17785  idffth  17828  ressffth  17833  catciso  18005
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