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Theorem comffval 17742
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 17741 . . 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 6910 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
9 comffval.x . . . . . . 7 (𝜑𝑋𝐵)
10 comffval.y . . . . . . 7 (𝜑𝑌𝐵)
11 op2ndg 8027 . . . . . . 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 2777 . . . 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 6910 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
18 df-ov 7434 . . . 4 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1917, 18eqtr4di 2795 . . 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 5724 . 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 8104 . . 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 1540  wcel 2108  Vcvv 3480  cop 4632   × cxp 5683  cfv 6561  (class class class)co 7431  cmpo 7433  2nd c2nd 8013  Basecbs 17247  Hom chom 17308  compcco 17309  compfccomf 17710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-comf 17714
This theorem is referenced by:  comfval  17743  comffval2  17745  comffn  17748
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