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Theorem comfffval 17435
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
Hypotheses
Ref Expression
comfffval.o 𝑂 = (compf𝐶)
comfffval.b 𝐵 = (Base‘𝐶)
comfffval.h 𝐻 = (Hom ‘𝐶)
comfffval.x · = (comp‘𝐶)
Assertion
Ref Expression
comfffval 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑓,𝑔,𝑥,𝑦,𝐶   · ,𝑓,𝑔,𝑥   𝑓,𝐻,𝑔,𝑥
Allowed substitution hints:   𝐵(𝑓,𝑔)   · (𝑦)   𝐻(𝑦)   𝑂(𝑥,𝑦,𝑓,𝑔)

Proof of Theorem comfffval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 comfffval.o . 2 𝑂 = (compf𝐶)
2 fveq2 6792 . . . . . . 7 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 comfffval.b . . . . . . 7 𝐵 = (Base‘𝐶)
42, 3eqtr4di 2791 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
54sqxpeqd 5623 . . . . 5 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
6 fveq2 6792 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
7 comfffval.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
86, 7eqtr4di 2791 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
98oveqd 7312 . . . . . 6 (𝑐 = 𝐶 → ((2nd𝑥)(Hom ‘𝑐)𝑦) = ((2nd𝑥)𝐻𝑦))
108fveq1d 6794 . . . . . 6 (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐻𝑥))
11 fveq2 6792 . . . . . . . . 9 (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
12 comfffval.x . . . . . . . . 9 · = (comp‘𝐶)
1311, 12eqtr4di 2791 . . . . . . . 8 (𝑐 = 𝐶 → (comp‘𝑐) = · )
1413oveqd 7312 . . . . . . 7 (𝑐 = 𝐶 → (𝑥(comp‘𝑐)𝑦) = (𝑥 · 𝑦))
1514oveqd 7312 . . . . . 6 (𝑐 = 𝐶 → (𝑔(𝑥(comp‘𝑐)𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
169, 10, 15mpoeq123dv 7370 . . . . 5 (𝑐 = 𝐶 → (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓)) = (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
175, 4, 16mpoeq123dv 7370 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
18 df-comf 17408 . . . 4 compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))))
193fvexi 6806 . . . . . 6 𝐵 ∈ V
2019, 19xpex 7623 . . . . 5 (𝐵 × 𝐵) ∈ V
2120, 19mpoex 7940 . . . 4 (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) ∈ V
2217, 18, 21fvmpt 6895 . . 3 (𝐶 ∈ V → (compf𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
23 fvprc 6784 . . . 4 𝐶 ∈ V → (compf𝐶) = ∅)
24 fvprc 6784 . . . . . . 7 𝐶 ∈ V → (Base‘𝐶) = ∅)
253, 24eqtrid 2785 . . . . . 6 𝐶 ∈ V → 𝐵 = ∅)
2625olcd 870 . . . . 5 𝐶 ∈ V → ((𝐵 × 𝐵) = ∅ ∨ 𝐵 = ∅))
27 0mpo0 7378 . . . . 5 (((𝐵 × 𝐵) = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = ∅)
2826, 27syl 17 . . . 4 𝐶 ∈ V → (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = ∅)
2923, 28eqtr4d 2776 . . 3 𝐶 ∈ V → (compf𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))))
3022, 29pm2.61i 182 . 2 (compf𝐶) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
311, 30eqtri 2761 1 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 843   = wceq 1537  wcel 2101  Vcvv 3434  c0 4259   × cxp 5589  cfv 6447  (class class class)co 7295  cmpo 7297  2nd c2nd 7850  Basecbs 16940  Hom chom 17001  compcco 17002  compfccomf 17404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-1st 7851  df-2nd 7852  df-comf 17408
This theorem is referenced by:  comffval  17436  comfffval2  17438  comfffn  17441  comfeq  17443
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