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Theorem comffval 16182
 Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval.o 𝑂 = (compf𝐶)
comfffval.b 𝐵 = (Base‘𝐶)
comfffval.h 𝐻 = (Hom ‘𝐶)
comfffval.x · = (comp‘𝐶)
comffval.x (𝜑𝑋𝐵)
comffval.y (𝜑𝑌𝐵)
comffval.z (𝜑𝑍𝐵)
Assertion
Ref Expression
comffval (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
Distinct variable groups:   𝑓,𝑔,𝐶   𝜑,𝑓,𝑔   · ,𝑓,𝑔   𝑓,𝑋,𝑔   𝑓,𝑌,𝑔   𝑓,𝑍,𝑔   𝑓,𝐻,𝑔
Allowed substitution hints:   𝐵(𝑓,𝑔)   𝑂(𝑓,𝑔)

Proof of Theorem comffval
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfffval.o . . . 4 𝑂 = (compf𝐶)
2 comfffval.b . . . 4 𝐵 = (Base‘𝐶)
3 comfffval.h . . . 4 𝐻 = (Hom ‘𝐶)
4 comfffval.x . . . 4 · = (comp‘𝐶)
51, 2, 3, 4comfffval 16181 . . 3 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑧), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓)))
65a1i 11 . 2 (𝜑𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑧), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓))))
7 simprl 790 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑥 = ⟨𝑋, 𝑌⟩)
87fveq2d 6107 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
9 comffval.x . . . . . . 7 (𝜑𝑋𝐵)
10 comffval.y . . . . . . 7 (𝜑𝑌𝐵)
11 op2ndg 7072 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
129, 10, 11syl2anc 691 . . . . . 6 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1312adantr 480 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
148, 13eqtrd 2644 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑥) = 𝑌)
15 simprr 792 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
1614, 15oveq12d 6567 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd𝑥)𝐻𝑧) = (𝑌𝐻𝑍))
177fveq2d 6107 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
18 df-ov 6552 . . . 4 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1917, 18syl6eqr 2662 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻𝑥) = (𝑋𝐻𝑌))
207, 15oveq12d 6567 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑥 · 𝑧) = (⟨𝑋, 𝑌· 𝑍))
2120oveqd 6566 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔(𝑥 · 𝑧)𝑓) = (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓))
2216, 19, 21mpt2eq123dv 6615 . 2 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd𝑥)𝐻𝑧), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓)) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
23 opelxpi 5072 . . 3 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
249, 10, 23syl2anc 691 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
25 comffval.z . 2 (𝜑𝑍𝐵)
26 ovex 6577 . . . 4 (𝑌𝐻𝑍) ∈ V
27 ovex 6577 . . . 4 (𝑋𝐻𝑌) ∈ V
2826, 27mpt2ex 7136 . . 3 (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)) ∈ V
2928a1i 11 . 2 (𝜑 → (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)) ∈ V)
306, 22, 24, 25, 29ovmpt2d 6686 1 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  compfccomf 16151 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-comf 16155 This theorem is referenced by:  comfval  16183  comffval2  16185  comffn  16188
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