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Theorem curf11 16689
 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 2610 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
10 eqid 2610 . . . 4 (Id‘𝐶) = (Id‘𝐶)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 16688 . . 3 (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
12 fvex 6113 . . . . . 6 (Base‘𝐷) ∈ V
136, 12eqeltri 2684 . . . . 5 𝐵 ∈ V
1413mptex 6390 . . . 4 (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)) ∈ V
1513, 13mpt2ex 7136 . . . 4 (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
1614, 15op1std 7069 . . 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)))
1711, 16syl 17 . 2 (𝜑 → (1st𝐾) = (𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)))
18 simpr 476 . . 3 ((𝜑𝑦 = 𝑌) → 𝑦 = 𝑌)
1918oveq2d 6565 . 2 ((𝜑𝑦 = 𝑌) → (𝑋(1st𝐹)𝑦) = (𝑋(1st𝐹)𝑌))
20 curf11.y . 2 (𝜑𝑌𝐵)
21 ovex 6577 . . 3 (𝑋(1st𝐹)𝑌) ∈ V
2221a1i 11 . 2 (𝜑 → (𝑋(1st𝐹)𝑌) ∈ V)
2317, 19, 20, 22fvmptd 6197 1 (𝜑 → ((1st𝐾)‘𝑌) = (𝑋(1st𝐹)𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  Catccat 16148  Idccid 16149   Func cfunc 16337   ×c cxpc 16631   curryF ccurf 16673 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-curf 16677 This theorem is referenced by:  curf1cl  16691  curf2cl  16694  curfcl  16695  uncfcurf  16702  diag11  16706  yon11  16727
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