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Theorem resf2nd 16907
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 6908 . 2 (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((2nd ‘(𝐹f 𝐻))‘⟨𝑋, 𝑌⟩)
2 resf1st.f . . . . . 6 (𝜑𝐹𝑉)
3 resf1st.h . . . . . 6 (𝜑𝐻𝑊)
42, 3resfval 16904 . . . . 5 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
54fveq2d 6437 . . . 4 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
6 fvex 6446 . . . . . 6 (1st𝐹) ∈ V
76resex 5680 . . . . 5 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
8 dmexg 7358 . . . . . 6 (𝐻𝑊 → dom 𝐻 ∈ V)
9 mptexg 6740 . . . . . 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 7441 . . . . 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 583 . . . 4 (𝜑 → (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
135, 12eqtrd 2861 . . 3 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
14 simpr 479 . . . . . 6 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
1514fveq2d 6437 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd𝐹)‘𝑧) = ((2nd𝐹)‘⟨𝑋, 𝑌⟩))
16 df-ov 6908 . . . . 5 (𝑋(2nd𝐹)𝑌) = ((2nd𝐹)‘⟨𝑋, 𝑌⟩)
1715, 16syl6eqr 2879 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd𝐹)‘𝑧) = (𝑋(2nd𝐹)𝑌))
1814fveq2d 6437 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
19 df-ov 6908 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2018, 19syl6eqr 2879 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝑋𝐻𝑌))
2117, 20reseq12d 5630 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
22 resf2nd.x . . . . 5 (𝜑𝑋𝑆)
23 resf2nd.y . . . . 5 (𝜑𝑌𝑆)
24 opelxpi 5379 . . . . 5 ((𝑋𝑆𝑌𝑆) → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
2522, 23, 24syl2anc 581 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
26 resf1st.s . . . . 5 (𝜑𝐻 Fn (𝑆 × 𝑆))
27 fndm 6223 . . . . 5 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
2826, 27syl 17 . . . 4 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
2925, 28eleqtrrd 2909 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
30 ovex 6937 . . . . 5 (𝑋(2nd𝐹)𝑌) ∈ V
3130resex 5680 . . . 4 ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V
3231a1i 11 . . 3 (𝜑 → ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V)
3313, 21, 29, 32fvmptd 6535 . 2 (𝜑 → ((2nd ‘(𝐹f 𝐻))‘⟨𝑋, 𝑌⟩) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
341, 33syl5eq 2873 1 (𝜑 → (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  Vcvv 3414  cop 4403  cmpt 4952   × cxp 5340  dom cdm 5342  cres 5344   Fn wfn 6118  cfv 6123  (class class class)co 6905  1st c1st 7426  2nd c2nd 7427  f cresf 16869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-2nd 7429  df-resf 16873
This theorem is referenced by:  funcres  16908
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