MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  idaval Structured version   Visualization version   GIF version

Theorem idaval 16477
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)
idaval.x (𝜑𝑋𝐵)
Assertion
Ref Expression
idaval (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)

Proof of Theorem idaval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 idafval.i . . 3 𝐼 = (Ida𝐶)
2 idafval.b . . 3 𝐵 = (Base‘𝐶)
3 idafval.c . . 3 (𝜑𝐶 ∈ Cat)
4 idafval.1 . . 3 1 = (Id‘𝐶)
51, 2, 3, 4idafval 16476 . 2 (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
6 simpr 475 . . 3 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
76fveq2d 6092 . . 3 ((𝜑𝑥 = 𝑋) → ( 1𝑥) = ( 1𝑋))
86, 6, 7oteq123d 4349 . 2 ((𝜑𝑥 = 𝑋) → ⟨𝑥, 𝑥, ( 1𝑥)⟩ = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
9 idaval.x . 2 (𝜑𝑋𝐵)
10 otex 4854 . . 3 𝑋, 𝑋, ( 1𝑋)⟩ ∈ V
1110a1i 11 . 2 (𝜑 → ⟨𝑋, 𝑋, ( 1𝑋)⟩ ∈ V)
125, 8, 9, 11fvmptd 6182 1 (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cotp 4132  cfv 5790  Basecbs 15641  Catccat 16094  Idccid 16095  Idacida 16472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
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 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-ot 4133  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ida 16474
This theorem is referenced by:  ida2  16478  idahom  16479
  Copyright terms: Public domain W3C validator