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Theorem resf2nd 16998
 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 7026 . 2 (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((2nd ‘(𝐹f 𝐻))‘⟨𝑋, 𝑌⟩)
2 resf1st.f . . . . . 6 (𝜑𝐹𝑉)
3 resf1st.h . . . . . 6 (𝜑𝐻𝑊)
42, 3resfval 16995 . . . . 5 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
54fveq2d 6549 . . . 4 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
6 fvex 6558 . . . . . 6 (1st𝐹) ∈ V
76resex 5787 . . . . 5 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
8 dmexg 7476 . . . . . 6 (𝐻𝑊 → dom 𝐻 ∈ V)
9 mptexg 6857 . . . . . 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 7565 . . . . 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 2833 . . 3 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
14 simpr 485 . . . . . 6 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
1514fveq2d 6549 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd𝐹)‘𝑧) = ((2nd𝐹)‘⟨𝑋, 𝑌⟩))
16 df-ov 7026 . . . . 5 (𝑋(2nd𝐹)𝑌) = ((2nd𝐹)‘⟨𝑋, 𝑌⟩)
1715, 16syl6eqr 2851 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd𝐹)‘𝑧) = (𝑋(2nd𝐹)𝑌))
1814fveq2d 6549 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
19 df-ov 7026 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2018, 19syl6eqr 2851 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝑋𝐻𝑌))
2117, 20reseq12d 5742 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
22 resf2nd.x . . . . 5 (𝜑𝑋𝑆)
23 resf2nd.y . . . . 5 (𝜑𝑌𝑆)
2422, 23opelxpd 5488 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
25 resf1st.s . . . . 5 (𝜑𝐻 Fn (𝑆 × 𝑆))
26 fndm 6332 . . . . 5 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
2725, 26syl 17 . . . 4 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
2824, 27eleqtrrd 2888 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
29 ovex 7055 . . . . 5 (𝑋(2nd𝐹)𝑌) ∈ V
3029resex 5787 . . . 4 ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V
3130a1i 11 . . 3 (𝜑 → ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V)
3213, 21, 28, 31fvmptd 6648 . 2 (𝜑 → ((2nd ‘(𝐹f 𝐻))‘⟨𝑋, 𝑌⟩) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
331, 32syl5eq 2845 1 (𝜑 → (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1525   ∈ wcel 2083  Vcvv 3440  ⟨cop 4484   ↦ cmpt 5047   × cxp 5448  dom cdm 5450   ↾ cres 5452   Fn wfn 6227  ‘cfv 6232  (class class class)co 7023  1st c1st 7550  2nd c2nd 7551   ↾f cresf 16960 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-2nd 7553  df-resf 16964 This theorem is referenced by:  funcres  16999
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