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Theorem ressress 16958
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 766 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘𝑊) ⊆ 𝐴)
2 simpr1 1193 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝑊 ∈ V)
3 simpr2 1194 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐴𝑋)
4 eqid 2738 . . . . . . . . . 10 (𝑊s 𝐴) = (𝑊s 𝐴)
5 eqid 2738 . . . . . . . . . 10 (Base‘𝑊) = (Base‘𝑊)
64, 5ressval2 16946 . . . . . . . . 9 ((¬ (Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑋) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
71, 2, 3, 6syl3anc 1370 . . . . . . . 8 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
8 inass 4153 . . . . . . . . . . 11 ((𝐴𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
9 in12 4154 . . . . . . . . . . 11 (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
108, 9eqtri 2766 . . . . . . . . . 10 ((𝐴𝐵) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
114, 5ressbas 16947 . . . . . . . . . . . 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 4146 . . . . . . . . . 10 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘(𝑊s 𝐴))))
1410, 13eqtr2id 2791 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘(𝑊s 𝐴))) = ((𝐴𝐵) ∩ (Base‘𝑊)))
1514opeq2d 4811 . . . . . . . 8 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊s 𝐴)))⟩ = ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩)
167, 15oveq12d 7293 . . . . . . 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 6787 . . . . . . . . 9 (Base‘𝑊) ∈ V
1817inex2 5242 . . . . . . . 8 ((𝐴𝐵) ∩ (Base‘𝑊)) ∈ V
19 setsabs 16880 . . . . . . . 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 2778 . . . . . 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 764 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵)
23 ovexd 7310 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) ∈ V)
24 simpr3 1195 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
25 eqid 2738 . . . . . . . 8 ((𝑊s 𝐴) ↾s 𝐵) = ((𝑊s 𝐴) ↾s 𝐵)
26 eqid 2738 . . . . . . . 8 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
2725, 26ressval2 16946 . . . . . . 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 1370 . . . . . 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 4162 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
30 sstr 3929 . . . . . . . . 9 (((Base‘𝑊) ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐴) → (Base‘𝑊) ⊆ 𝐴)
3129, 30mpan2 688 . . . . . . . 8 ((Base‘𝑊) ⊆ (𝐴𝐵) → (Base‘𝑊) ⊆ 𝐴)
321, 31nsyl 140 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘𝑊) ⊆ (𝐴𝐵))
33 inex1g 5243 . . . . . . . 8 (𝐴𝑋 → (𝐴𝐵) ∈ V)
343, 33syl 17 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴𝐵) ∈ V)
35 eqid 2738 . . . . . . . 8 (𝑊s (𝐴𝐵)) = (𝑊s (𝐴𝐵))
3635, 5ressval2 16946 . . . . . . 7 ((¬ (Base‘𝑊) ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
3732, 2, 34, 36syl3anc 1370 . . . . . 6 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
3821, 28, 373eqtr4d 2788 . . . . 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 7308 . . . . . . . 8 (𝑊s 𝐴) ∈ V
4125, 26ressid2 16945 . . . . . . . 8 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊s 𝐴) ∈ V ∧ 𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
4240, 41mp3an2 1448 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
43423ad2antr3 1189 . . . . . 6 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
44 in32 4155 . . . . . . . . 9 ((𝐴𝐵) ∩ (Base‘𝑊)) = ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵)
45 simpr2 1194 . . . . . . . . . . . 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 3959 . . . . . . . . . 10 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵)
49 df-ss 3904 . . . . . . . . . 10 ((𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵 ↔ ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
5048, 49sylib 217 . . . . . . . . 9 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
5144, 50eqtr2id 2791 . . . . . . . 8 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) = ((𝐴𝐵) ∩ (Base‘𝑊)))
5251oveq2d 7291 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴 ∩ (Base‘𝑊))) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
535ressinbas 16955 . . . . . . . 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 16955 . . . . . . . 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 2788 . . . . . 6 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
5843, 57eqtrd 2778 . . . . 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 16945 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑋) → (𝑊s 𝐴) = 𝑊)
61603adant3r3 1183 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = 𝑊)
6261oveq1d 7290 . . . . . 6 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐵))
63 inss2 4163 . . . . . . . . . . 11 (𝐵 ∩ (Base‘𝑊)) ⊆ (Base‘𝑊)
64 simpl 483 . . . . . . . . . . 11 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (Base‘𝑊) ⊆ 𝐴)
6563, 64sstrid 3932 . . . . . . . . . 10 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴)
66 sseqin2 4149 . . . . . . . . . 10 ((𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
6765, 66sylib 217 . . . . . . . . 9 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
688, 67eqtr2id 2791 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘𝑊)) = ((𝐴𝐵) ∩ (Base‘𝑊)))
6968oveq2d 7291 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐵 ∩ (Base‘𝑊))) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
70 simpr3 1195 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
715ressinbas 16955 . . . . . . . 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 1194 . . . . . . . 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 2788 . . . . . 6 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐵) = (𝑊s (𝐴𝐵)))
7662, 75eqtrd 2778 . . . . 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 183 . . 3 ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
79783expib 1121 . 2 (𝑊 ∈ V → ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
80 ress0 16953 . . . 4 (∅ ↾s 𝐵) = ∅
81 reldmress 16943 . . . . . 6 Rel dom ↾s
8281ovprc1 7314 . . . . 5 𝑊 ∈ V → (𝑊s 𝐴) = ∅)
8382oveq1d 7290 . . . 4 𝑊 ∈ V → ((𝑊s 𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
8481ovprc1 7314 . . . 4 𝑊 ∈ V → (𝑊s (𝐴𝐵)) = ∅)
8580, 83, 843eqtr4a 2804 . . 3 𝑊 ∈ V → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
8685a1d 25 . 2 𝑊 ∈ V → ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
8779, 86pm2.61i 182 1 ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3432  cin 3886  wss 3887  c0 4256  cop 4567  cfv 6433  (class class class)co 7275   sSet csts 16864  ndxcnx 16894  Basecbs 16912  s cress 16941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-1cn 10929  ax-addcl 10931
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-nn 11974  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942
This theorem is referenced by:  ressabs  16959  xrge00  31295  xrge0slmod  31548  fldexttr  31733  esumpfinvallem  32042  lmhmlnmsplit  40912
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