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Theorem ressress 16213
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 785 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘𝑊) ⊆ 𝐴)
2 simpr1 1248 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝑊 ∈ V)
3 simpr2 1250 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐴𝑋)
4 eqid 2765 . . . . . . . . . 10 (𝑊s 𝐴) = (𝑊s 𝐴)
5 eqid 2765 . . . . . . . . . 10 (Base‘𝑊) = (Base‘𝑊)
64, 5ressval2 16203 . . . . . . . . 9 ((¬ (Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑋) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
71, 2, 3, 6syl3anc 1490 . . . . . . . 8 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
8 inass 3983 . . . . . . . . . . 11 ((𝐴𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
9 in12 3984 . . . . . . . . . . 11 (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
108, 9eqtri 2787 . . . . . . . . . 10 ((𝐴𝐵) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
114, 5ressbas 16204 . . . . . . . . . . . 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 3976 . . . . . . . . . 10 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘(𝑊s 𝐴))))
1410, 13syl5req 2812 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘(𝑊s 𝐴))) = ((𝐴𝐵) ∩ (Base‘𝑊)))
1514opeq2d 4566 . . . . . . . 8 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊s 𝐴)))⟩ = ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩)
167, 15oveq12d 6860 . . . . . . 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 6388 . . . . . . . . 9 (Base‘𝑊) ∈ V
1817inex2 4961 . . . . . . . 8 ((𝐴𝐵) ∩ (Base‘𝑊)) ∈ V
19 setsabs 16176 . . . . . . . 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 580 . . . . . . 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 2799 . . . . . 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 783 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵)
23 ovexd 6876 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) ∈ V)
24 simpr3 1252 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
25 eqid 2765 . . . . . . . 8 ((𝑊s 𝐴) ↾s 𝐵) = ((𝑊s 𝐴) ↾s 𝐵)
26 eqid 2765 . . . . . . . 8 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
2725, 26ressval2 16203 . . . . . . 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 1490 . . . . . 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 3992 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
30 sstr 3769 . . . . . . . . 9 (((Base‘𝑊) ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐴) → (Base‘𝑊) ⊆ 𝐴)
3129, 30mpan2 682 . . . . . . . 8 ((Base‘𝑊) ⊆ (𝐴𝐵) → (Base‘𝑊) ⊆ 𝐴)
321, 31nsyl 137 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘𝑊) ⊆ (𝐴𝐵))
33 inex1g 4962 . . . . . . . 8 (𝐴𝑋 → (𝐴𝐵) ∈ V)
343, 33syl 17 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴𝐵) ∈ V)
35 eqid 2765 . . . . . . . 8 (𝑊s (𝐴𝐵)) = (𝑊s (𝐴𝐵))
3635, 5ressval2 16203 . . . . . . 7 ((¬ (Base‘𝑊) ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
3732, 2, 34, 36syl3anc 1490 . . . . . 6 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
3821, 28, 373eqtr4d 2809 . . . . 5 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
3938exp31 410 . . . 4 (¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 → (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))))
40 ovex 6874 . . . . . . . 8 (𝑊s 𝐴) ∈ V
4125, 26ressid2 16202 . . . . . . . 8 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊s 𝐴) ∈ V ∧ 𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
4240, 41mp3an2 1573 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
43423ad2antr3 1241 . . . . . 6 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
44 in32 3985 . . . . . . . . 9 ((𝐴𝐵) ∩ (Base‘𝑊)) = ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵)
45 simpr2 1250 . . . . . . . . . . . 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 474 . . . . . . . . . . 11 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (Base‘(𝑊s 𝐴)) ⊆ 𝐵)
4846, 47eqsstrd 3799 . . . . . . . . . 10 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵)
49 df-ss 3746 . . . . . . . . . 10 ((𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵 ↔ ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
5048, 49sylib 209 . . . . . . . . 9 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
5144, 50syl5req 2812 . . . . . . . 8 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) = ((𝐴𝐵) ∩ (Base‘𝑊)))
5251oveq2d 6858 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴 ∩ (Base‘𝑊))) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
535ressinbas 16210 . . . . . . . 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 16210 . . . . . . . 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 2809 . . . . . 6 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
5843, 57eqtrd 2799 . . . . 5 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
5958ex 401 . . . 4 ((Base‘(𝑊s 𝐴)) ⊆ 𝐵 → ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
604, 5ressid2 16202 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑋) → (𝑊s 𝐴) = 𝑊)
61603adant3r3 1235 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = 𝑊)
6261oveq1d 6857 . . . . . 6 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐵))
63 inss2 3993 . . . . . . . . . . 11 (𝐵 ∩ (Base‘𝑊)) ⊆ (Base‘𝑊)
64 simpl 474 . . . . . . . . . . 11 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (Base‘𝑊) ⊆ 𝐴)
6563, 64syl5ss 3772 . . . . . . . . . 10 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴)
66 sseqin2 3979 . . . . . . . . . 10 ((𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
6765, 66sylib 209 . . . . . . . . 9 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
688, 67syl5req 2812 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘𝑊)) = ((𝐴𝐵) ∩ (Base‘𝑊)))
6968oveq2d 6858 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐵 ∩ (Base‘𝑊))) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
70 simpr3 1252 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
715ressinbas 16210 . . . . . . . 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 1250 . . . . . . . 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 2809 . . . . . 6 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐵) = (𝑊s (𝐴𝐵)))
7662, 75eqtrd 2799 . . . . 5 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
7776ex 401 . . . 4 ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
7839, 59, 77pm2.61ii 177 . . 3 ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
79783expib 1152 . 2 (𝑊 ∈ V → ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
80 ress0 16208 . . . 4 (∅ ↾s 𝐵) = ∅
81 reldmress 16200 . . . . . 6 Rel dom ↾s
8281ovprc1 6880 . . . . 5 𝑊 ∈ V → (𝑊s 𝐴) = ∅)
8382oveq1d 6857 . . . 4 𝑊 ∈ V → ((𝑊s 𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
8481ovprc1 6880 . . . 4 𝑊 ∈ V → (𝑊s (𝐴𝐵)) = ∅)
8580, 83, 843eqtr4a 2825 . . 3 𝑊 ∈ V → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
8685a1d 25 . 2 𝑊 ∈ V → ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
8779, 86pm2.61i 176 1 ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  Vcvv 3350  cin 3731  wss 3732  c0 4079  cop 4340  cfv 6068  (class class class)co 6842  ndxcnx 16129   sSet csts 16130  Basecbs 16132  s cress 16133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-1cn 10247  ax-addcl 10249
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-nn 11275  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140
This theorem is referenced by:  ressabs  16214  xrge00  30068  xrge0slmod  30226  esumpfinvallem  30518  lmhmlnmsplit  38266
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