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Theorem resf2nd 17946
Description: Value of the functor restriction operator on morphisms. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resf1st.f (𝜑𝐹𝑉)
resf1st.h (𝜑𝐻𝑊)
resf1st.s (𝜑𝐻 Fn (𝑆 × 𝑆))
resf2nd.x (𝜑𝑋𝑆)
resf2nd.y (𝜑𝑌𝑆)
Assertion
Ref Expression
resf2nd (𝜑 → (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))

Proof of Theorem resf2nd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ov 7434 . 2 (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((2nd ‘(𝐹f 𝐻))‘⟨𝑋, 𝑌⟩)
2 resf1st.f . . . . . 6 (𝜑𝐹𝑉)
3 resf1st.h . . . . . 6 (𝜑𝐻𝑊)
42, 3resfval 17943 . . . . 5 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
54fveq2d 6911 . . . 4 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
6 fvex 6920 . . . . . 6 (1st𝐹) ∈ V
76resex 6049 . . . . 5 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
8 dmexg 7924 . . . . . 6 (𝐻𝑊 → dom 𝐻 ∈ V)
9 mptexg 7241 . . . . . 6 (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V)
103, 8, 93syl 18 . . . . 5 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V)
11 op2ndg 8026 . . . . 5 ((((1st𝐹) ↾ dom dom 𝐻) ∈ V ∧ (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V) → (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
127, 10, 11sylancr 587 . . . 4 (𝜑 → (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
135, 12eqtrd 2775 . . 3 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
14 simpr 484 . . . . . 6 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
1514fveq2d 6911 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd𝐹)‘𝑧) = ((2nd𝐹)‘⟨𝑋, 𝑌⟩))
16 df-ov 7434 . . . . 5 (𝑋(2nd𝐹)𝑌) = ((2nd𝐹)‘⟨𝑋, 𝑌⟩)
1715, 16eqtr4di 2793 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd𝐹)‘𝑧) = (𝑋(2nd𝐹)𝑌))
1814fveq2d 6911 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
19 df-ov 7434 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2018, 19eqtr4di 2793 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝑋𝐻𝑌))
2117, 20reseq12d 6001 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
22 resf2nd.x . . . . 5 (𝜑𝑋𝑆)
23 resf2nd.y . . . . 5 (𝜑𝑌𝑆)
2422, 23opelxpd 5728 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
25 resf1st.s . . . . 5 (𝜑𝐻 Fn (𝑆 × 𝑆))
2625fndmd 6674 . . . 4 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
2724, 26eleqtrrd 2842 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
28 ovex 7464 . . . . 5 (𝑋(2nd𝐹)𝑌) ∈ V
2928resex 6049 . . . 4 ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V
3029a1i 11 . . 3 (𝜑 → ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V)
3113, 21, 27, 30fvmptd 7023 . 2 (𝜑 → ((2nd ‘(𝐹f 𝐻))‘⟨𝑋, 𝑌⟩) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
321, 31eqtrid 2787 1 (𝜑 → (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cmpt 5231   × cxp 5687  dom cdm 5689  cres 5691   Fn wfn 6558  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  f cresf 17908
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-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-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-2nd 8014  df-resf 17912
This theorem is referenced by:  funcres  17947
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