MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ressress Structured version   Visualization version   GIF version

Theorem ressress 16557
Description: Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
ressress ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))

Proof of Theorem ressress
StepHypRef Expression
1 simplr 765 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘𝑊) ⊆ 𝐴)
2 simpr1 1188 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝑊 ∈ V)
3 simpr2 1189 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐴𝑋)
4 eqid 2826 . . . . . . . . . 10 (𝑊s 𝐴) = (𝑊s 𝐴)
5 eqid 2826 . . . . . . . . . 10 (Base‘𝑊) = (Base‘𝑊)
64, 5ressval2 16548 . . . . . . . . 9 ((¬ (Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑋) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
71, 2, 3, 6syl3anc 1365 . . . . . . . 8 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
8 inass 4200 . . . . . . . . . . 11 ((𝐴𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
9 in12 4201 . . . . . . . . . . 11 (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
108, 9eqtri 2849 . . . . . . . . . 10 ((𝐴𝐵) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
114, 5ressbas 16549 . . . . . . . . . . . 12 (𝐴𝑋 → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊s 𝐴)))
123, 11syl 17 . . . . . . . . . . 11 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊s 𝐴)))
1312ineq2d 4193 . . . . . . . . . 10 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘(𝑊s 𝐴))))
1410, 13syl5req 2874 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘(𝑊s 𝐴))) = ((𝐴𝐵) ∩ (Base‘𝑊)))
1514opeq2d 4809 . . . . . . . 8 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊s 𝐴)))⟩ = ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩)
167, 15oveq12d 7168 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊s 𝐴)))⟩) = ((𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
17 fvex 6682 . . . . . . . . 9 (Base‘𝑊) ∈ V
1817inex2 5219 . . . . . . . 8 ((𝐴𝐵) ∩ (Base‘𝑊)) ∈ V
19 setsabs 16521 . . . . . . . 8 ((𝑊 ∈ V ∧ ((𝐴𝐵) ∩ (Base‘𝑊)) ∈ V) → ((𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
202, 18, 19sylancl 586 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
2116, 20eqtrd 2861 . . . . . 6 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊s 𝐴)))⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
22 simpll 763 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵)
23 ovexd 7185 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) ∈ V)
24 simpr3 1190 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
25 eqid 2826 . . . . . . . 8 ((𝑊s 𝐴) ↾s 𝐵) = ((𝑊s 𝐴) ↾s 𝐵)
26 eqid 2826 . . . . . . . 8 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
2725, 26ressval2 16548 . . . . . . 7 ((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊s 𝐴) ∈ V ∧ 𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = ((𝑊s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊s 𝐴)))⟩))
2822, 23, 24, 27syl3anc 1365 . . . . . 6 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = ((𝑊s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊s 𝐴)))⟩))
29 inss1 4209 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
30 sstr 3979 . . . . . . . . 9 (((Base‘𝑊) ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐴) → (Base‘𝑊) ⊆ 𝐴)
3129, 30mpan2 687 . . . . . . . 8 ((Base‘𝑊) ⊆ (𝐴𝐵) → (Base‘𝑊) ⊆ 𝐴)
321, 31nsyl 142 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘𝑊) ⊆ (𝐴𝐵))
33 inex1g 5220 . . . . . . . 8 (𝐴𝑋 → (𝐴𝐵) ∈ V)
343, 33syl 17 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴𝐵) ∈ V)
35 eqid 2826 . . . . . . . 8 (𝑊s (𝐴𝐵)) = (𝑊s (𝐴𝐵))
3635, 5ressval2 16548 . . . . . . 7 ((¬ (Base‘𝑊) ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
3732, 2, 34, 36syl3anc 1365 . . . . . 6 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
3821, 28, 373eqtr4d 2871 . . . . 5 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
3938exp31 420 . . . 4 (¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 → (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))))
40 ovex 7183 . . . . . . . 8 (𝑊s 𝐴) ∈ V
4125, 26ressid2 16547 . . . . . . . 8 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊s 𝐴) ∈ V ∧ 𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
4240, 41mp3an2 1442 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
43423ad2antr3 1184 . . . . . 6 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
44 in32 4202 . . . . . . . . 9 ((𝐴𝐵) ∩ (Base‘𝑊)) = ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵)
45 simpr2 1189 . . . . . . . . . . . 12 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐴𝑋)
4645, 11syl 17 . . . . . . . . . . 11 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊s 𝐴)))
47 simpl 483 . . . . . . . . . . 11 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (Base‘(𝑊s 𝐴)) ⊆ 𝐵)
4846, 47eqsstrd 4009 . . . . . . . . . 10 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵)
49 df-ss 3956 . . . . . . . . . 10 ((𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵 ↔ ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
5048, 49sylib 219 . . . . . . . . 9 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
5144, 50syl5req 2874 . . . . . . . 8 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) = ((𝐴𝐵) ∩ (Base‘𝑊)))
5251oveq2d 7166 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴 ∩ (Base‘𝑊))) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
535ressinbas 16555 . . . . . . . 8 (𝐴𝑋 → (𝑊s 𝐴) = (𝑊s (𝐴 ∩ (Base‘𝑊))))
5445, 53syl 17 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = (𝑊s (𝐴 ∩ (Base‘𝑊))))
555ressinbas 16555 . . . . . . . 8 ((𝐴𝐵) ∈ V → (𝑊s (𝐴𝐵)) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
5645, 33, 553syl 18 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴𝐵)) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
5752, 54, 563eqtr4d 2871 . . . . . 6 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
5843, 57eqtrd 2861 . . . . 5 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
5958ex 413 . . . 4 ((Base‘(𝑊s 𝐴)) ⊆ 𝐵 → ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
604, 5ressid2 16547 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑋) → (𝑊s 𝐴) = 𝑊)
61603adant3r3 1178 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = 𝑊)
6261oveq1d 7165 . . . . . 6 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐵))
63 inss2 4210 . . . . . . . . . . 11 (𝐵 ∩ (Base‘𝑊)) ⊆ (Base‘𝑊)
64 simpl 483 . . . . . . . . . . 11 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (Base‘𝑊) ⊆ 𝐴)
6563, 64sstrid 3982 . . . . . . . . . 10 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴)
66 sseqin2 4196 . . . . . . . . . 10 ((𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
6765, 66sylib 219 . . . . . . . . 9 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
688, 67syl5req 2874 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘𝑊)) = ((𝐴𝐵) ∩ (Base‘𝑊)))
6968oveq2d 7166 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐵 ∩ (Base‘𝑊))) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
70 simpr3 1190 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
715ressinbas 16555 . . . . . . . 8 (𝐵𝑌 → (𝑊s 𝐵) = (𝑊s (𝐵 ∩ (Base‘𝑊))))
7270, 71syl 17 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐵) = (𝑊s (𝐵 ∩ (Base‘𝑊))))
73 simpr2 1189 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐴𝑋)
7473, 33, 553syl 18 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴𝐵)) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
7569, 72, 743eqtr4d 2871 . . . . . 6 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐵) = (𝑊s (𝐴𝐵)))
7662, 75eqtrd 2861 . . . . 5 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
7776ex 413 . . . 4 ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
7839, 59, 77pm2.61ii 184 . . 3 ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
79783expib 1116 . 2 (𝑊 ∈ V → ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
80 ress0 16553 . . . 4 (∅ ↾s 𝐵) = ∅
81 reldmress 16545 . . . . . 6 Rel dom ↾s
8281ovprc1 7189 . . . . 5 𝑊 ∈ V → (𝑊s 𝐴) = ∅)
8382oveq1d 7165 . . . 4 𝑊 ∈ V → ((𝑊s 𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
8481ovprc1 7189 . . . 4 𝑊 ∈ V → (𝑊s (𝐴𝐵)) = ∅)
8580, 83, 843eqtr4a 2887 . . 3 𝑊 ∈ V → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
8685a1d 25 . 2 𝑊 ∈ V → ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
8779, 86pm2.61i 183 1 ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  Vcvv 3500  cin 3939  wss 3940  c0 4295  cop 4570  cfv 6354  (class class class)co 7150  ndxcnx 16475   sSet csts 16476  Basecbs 16478  s cress 16479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-1cn 10589  ax-addcl 10591
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-nn 11633  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486
This theorem is referenced by:  ressabs  16558  xrge00  30606  xrge0slmod  30850  fldexttr  30953  esumpfinvallem  31238  lmhmlnmsplit  39571
  Copyright terms: Public domain W3C validator