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Theorem comffval 17744
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 17743 . . 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 771 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑥 = ⟨𝑋, 𝑌⟩)
87fveq2d 6911 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
9 comffval.x . . . . . . 7 (𝜑𝑋𝐵)
10 comffval.y . . . . . . 7 (𝜑𝑌𝐵)
11 op2ndg 8026 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
129, 10, 11syl2anc 584 . . . . . 6 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1312adantr 480 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
148, 13eqtrd 2775 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑥) = 𝑌)
15 simprr 773 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
1614, 15oveq12d 7449 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd𝑥)𝐻𝑧) = (𝑌𝐻𝑍))
177fveq2d 6911 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
18 df-ov 7434 . . . 4 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1917, 18eqtr4di 2793 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻𝑥) = (𝑋𝐻𝑌))
207, 15oveq12d 7449 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑥 · 𝑧) = (⟨𝑋, 𝑌· 𝑍))
2120oveqd 7448 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔(𝑥 · 𝑧)𝑓) = (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓))
2216, 19, 21mpoeq123dv 7508 . 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)))
239, 10opelxpd 5728 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
24 comffval.z . 2 (𝜑𝑍𝐵)
25 ovex 7464 . . . 4 (𝑌𝐻𝑍) ∈ V
26 ovex 7464 . . . 4 (𝑋𝐻𝑌) ∈ V
2725, 26mpoex 8103 . . 3 (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)) ∈ V
2827a1i 11 . 2 (𝜑 → (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)) ∈ V)
296, 22, 23, 24, 28ovmpod 7585 1 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637   × cxp 5687  cfv 6563  (class class class)co 7431  cmpo 7433  2nd c2nd 8012  Basecbs 17245  Hom chom 17309  compcco 17310  compfccomf 17712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-comf 17716
This theorem is referenced by:  comfval  17745  comffval2  17747  comffn  17750
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