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Theorem idafval 16472
Description: Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
idafval.i 𝐼 = (Ida𝐶)
idafval.b 𝐵 = (Base‘𝐶)
idafval.c (𝜑𝐶 ∈ Cat)
idafval.1 1 = (Id‘𝐶)
Assertion
Ref Expression
idafval (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
Distinct variable groups:   𝑥, 1   𝑥,𝐵   𝑥,𝐶   𝑥,𝐼   𝜑,𝑥

Proof of Theorem idafval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 idafval.i . 2 𝐼 = (Ida𝐶)
2 idafval.c . . 3 (𝜑𝐶 ∈ Cat)
3 fveq2 6084 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 idafval.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4syl6eqr 2657 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fveq2 6084 . . . . . . . 8 (𝑐 = 𝐶 → (Id‘𝑐) = (Id‘𝐶))
7 idafval.1 . . . . . . . 8 1 = (Id‘𝐶)
86, 7syl6eqr 2657 . . . . . . 7 (𝑐 = 𝐶 → (Id‘𝑐) = 1 )
98fveq1d 6086 . . . . . 6 (𝑐 = 𝐶 → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
109oteq3d 4344 . . . . 5 (𝑐 = 𝐶 → ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩ = ⟨𝑥, 𝑥, ( 1𝑥)⟩)
115, 10mpteq12dv 4653 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩) = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
12 df-ida 16470 . . . 4 Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
13 fvex 6094 . . . . . 6 (Base‘𝐶) ∈ V
144, 13eqeltri 2679 . . . . 5 𝐵 ∈ V
1514mptex 6364 . . . 4 (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩) ∈ V
1611, 12, 15fvmpt 6172 . . 3 (𝐶 ∈ Cat → (Ida𝐶) = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
172, 16syl 17 . 2 (𝜑 → (Ida𝐶) = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
181, 17syl5eq 2651 1 (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  Vcvv 3168  cotp 4128  cmpt 4633  cfv 5786  Basecbs 15637  Catccat 16090  Idccid 16091  Idacida 16468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-ot 4129  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ida 16470
This theorem is referenced by:  idaval  16473  idaf  16478
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