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Theorem prcofelvv 49359
Description: The pre-composition functor is an ordered pair. (Contributed by Zhi Wang, 4-Nov-2025.)
Hypotheses
Ref Expression
prcofelvv.f (𝜑𝐹𝑈)
prcofelvv.p (𝜑𝑃𝑉)
Assertion
Ref Expression
prcofelvv (𝜑 → (𝑃 −∘F 𝐹) ∈ (V × V))

Proof of Theorem prcofelvv
Dummy variables 𝑎 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2730 . . 3 ((1st𝑃) Func (2nd𝑃)) = ((1st𝑃) Func (2nd𝑃))
2 eqid 2730 . . 3 ((1st𝑃) Nat (2nd𝑃)) = ((1st𝑃) Nat (2nd𝑃))
3 prcofelvv.f . . 3 (𝜑𝐹𝑈)
4 prcofelvv.p . . 3 (𝜑𝑃𝑉)
5 eqidd 2731 . . 3 (𝜑 → (1st𝑃) = (1st𝑃))
6 eqidd 2731 . . 3 (𝜑 → (2nd𝑃) = (2nd𝑃))
71, 2, 3, 4, 5, 6prcofvalg 49355 . 2 (𝜑 → (𝑃 −∘F 𝐹) = ⟨(𝑘 ∈ ((1st𝑃) Func (2nd𝑃)) ↦ (𝑘func 𝐹)), (𝑘 ∈ ((1st𝑃) Func (2nd𝑃)), 𝑙 ∈ ((1st𝑃) Func (2nd𝑃)) ↦ (𝑎 ∈ (𝑘((1st𝑃) Nat (2nd𝑃))𝑙) ↦ (𝑎 ∘ (1st𝐹))))⟩)
8 ovex 7422 . . . 4 ((1st𝑃) Func (2nd𝑃)) ∈ V
98mptex 7199 . . 3 (𝑘 ∈ ((1st𝑃) Func (2nd𝑃)) ↦ (𝑘func 𝐹)) ∈ V
108, 8mpoex 8060 . . 3 (𝑘 ∈ ((1st𝑃) Func (2nd𝑃)), 𝑙 ∈ ((1st𝑃) Func (2nd𝑃)) ↦ (𝑎 ∈ (𝑘((1st𝑃) Nat (2nd𝑃))𝑙) ↦ (𝑎 ∘ (1st𝐹)))) ∈ V
119, 10opelvv 5680 . 2 ⟨(𝑘 ∈ ((1st𝑃) Func (2nd𝑃)) ↦ (𝑘func 𝐹)), (𝑘 ∈ ((1st𝑃) Func (2nd𝑃)), 𝑙 ∈ ((1st𝑃) Func (2nd𝑃)) ↦ (𝑎 ∈ (𝑘((1st𝑃) Nat (2nd𝑃))𝑙) ↦ (𝑎 ∘ (1st𝐹))))⟩ ∈ (V × V)
127, 11eqeltrdi 2837 1 (𝜑 → (𝑃 −∘F 𝐹) ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2109  Vcvv 3450  cop 4597  cmpt 5190   × cxp 5638  ccom 5644  cfv 6513  (class class class)co 7389  cmpo 7391  1st c1st 7968  2nd c2nd 7969   Func cfunc 17822  func ccofu 17824   Nat cnat 17912   −∘F cprcof 49352
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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-prcof 49353
This theorem is referenced by:  relran  49603  ranval3  49610  ranrcl4lem  49617  ranup  49621
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