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Theorem comffval 16340
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 16339 . . 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 793 . . . . . 6 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑥 = ⟨𝑋, 𝑌⟩)
87fveq2d 6182 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
9 comffval.x . . . . . . 7 (𝜑𝑋𝐵)
10 comffval.y . . . . . . 7 (𝜑𝑌𝐵)
11 op2ndg 7166 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
129, 10, 11syl2anc 692 . . . . . 6 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1312adantr 481 . . . . 5 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
148, 13eqtrd 2654 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑥) = 𝑌)
15 simprr 795 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
1614, 15oveq12d 6653 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd𝑥)𝐻𝑧) = (𝑌𝐻𝑍))
177fveq2d 6182 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
18 df-ov 6638 . . . 4 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1917, 18syl6eqr 2672 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝐻𝑥) = (𝑋𝐻𝑌))
207, 15oveq12d 6653 . . . 4 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑥 · 𝑧) = (⟨𝑋, 𝑌· 𝑍))
2120oveqd 6652 . . 3 ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔(𝑥 · 𝑧)𝑓) = (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓))
2216, 19, 21mpt2eq123dv 6702 . 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 5138 . . 3 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
249, 10, 23syl2anc 692 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
25 comffval.z . 2 (𝜑𝑍𝐵)
26 ovex 6663 . . . 4 (𝑌𝐻𝑍) ∈ V
27 ovex 6663 . . . 4 (𝑋𝐻𝑌) ∈ V
2826, 27mpt2ex 7232 . . 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 6773 1 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cop 4174   × cxp 5102  cfv 5876  (class class class)co 6635  cmpt2 6637  2nd c2nd 7152  Basecbs 15838  Hom chom 15933  compcco 15934  compfccomf 16309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-comf 16313
This theorem is referenced by:  comfval  16341  comffval2  16343  comffn  16346
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