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Theorem curf11 18218
Description: Value of the double evaluated curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
curfval.g 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curfval.a 𝐴 = (Base‘𝐶)
curfval.c (𝜑𝐶 ∈ Cat)
curfval.d (𝜑𝐷 ∈ Cat)
curfval.f (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curfval.b 𝐵 = (Base‘𝐷)
curf1.x (𝜑𝑋𝐴)
curf1.k 𝐾 = ((1st𝐺)‘𝑋)
curf11.y (𝜑𝑌𝐵)
Assertion
Ref Expression
curf11 (𝜑 → ((1st𝐾)‘𝑌) = (𝑋(1st𝐹)𝑌))

Proof of Theorem curf11
Dummy variables 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curfval.g . . . 4 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2 curfval.a . . . 4 𝐴 = (Base‘𝐶)
3 curfval.c . . . 4 (𝜑𝐶 ∈ Cat)
4 curfval.d . . . 4 (𝜑𝐷 ∈ Cat)
5 curfval.f . . . 4 (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6 curfval.b . . . 4 𝐵 = (Base‘𝐷)
7 curf1.x . . . 4 (𝜑𝑋𝐴)
8 curf1.k . . . 4 𝐾 = ((1st𝐺)‘𝑋)
9 eqid 2728 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
10 eqid 2728 . . . 4 (Id‘𝐶) = (Id‘𝐶)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 18217 . . 3 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
126fvexi 6911 . . . . 5 𝐵 ∈ V
1312mptex 7235 . . . 4 (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)) ∈ V
1412, 12mpoex 8084 . . . 4 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
1513, 14op1std 8003 . . 3 (𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ → (1st𝐾) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)))
1611, 15syl 17 . 2 (𝜑 → (1st𝐾) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)))
17 simpr 484 . . 3 ((𝜑𝑦 = 𝑌) → 𝑦 = 𝑌)
1817oveq2d 7436 . 2 ((𝜑𝑦 = 𝑌) → (𝑋(1st𝐹)𝑦) = (𝑋(1st𝐹)𝑌))
19 curf11.y . 2 (𝜑𝑌𝐵)
20 ovexd 7455 . 2 (𝜑 → (𝑋(1st𝐹)𝑌) ∈ V)
2116, 18, 19, 20fvmptd 7012 1 (𝜑 → ((1st𝐾)‘𝑌) = (𝑋(1st𝐹)𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  Vcvv 3471  cop 4635  cmpt 5231  cfv 6548  (class class class)co 7420  cmpo 7422  1st c1st 7991  2nd c2nd 7992  Basecbs 17180  Hom chom 17244  Catccat 17644  Idccid 17645   Func cfunc 17840   ×c cxpc 18159   curryF ccurf 18202
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 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-curf 18206
This theorem is referenced by:  curf1cl  18220  curf2cl  18223  curfcl  18224  uncfcurf  18231  diag11  18235  yon11  18256
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