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Theorem ressress 15870
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 791 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘𝑊) ⊆ 𝐴)
2 simpr1 1065 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝑊 ∈ V)
3 simpr2 1066 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐴𝑋)
4 eqid 2621 . . . . . . . . . 10 (𝑊s 𝐴) = (𝑊s 𝐴)
5 eqid 2621 . . . . . . . . . 10 (Base‘𝑊) = (Base‘𝑊)
64, 5ressval2 15861 . . . . . . . . 9 ((¬ (Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑋) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
71, 2, 3, 6syl3anc 1323 . . . . . . . 8 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
8 inass 3806 . . . . . . . . . . 11 ((𝐴𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
9 in12 3807 . . . . . . . . . . 11 (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
108, 9eqtri 2643 . . . . . . . . . 10 ((𝐴𝐵) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
114, 5ressbas 15862 . . . . . . . . . . . 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 3797 . . . . . . . . . 10 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘(𝑊s 𝐴))))
1410, 13syl5req 2668 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘(𝑊s 𝐴))) = ((𝐴𝐵) ∩ (Base‘𝑊)))
1514opeq2d 4382 . . . . . . . 8 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊s 𝐴)))⟩ = ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩)
167, 15oveq12d 6628 . . . . . . 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 6163 . . . . . . . . 9 (Base‘𝑊) ∈ V
1817inex2 4765 . . . . . . . 8 ((𝐴𝐵) ∩ (Base‘𝑊)) ∈ V
19 setsabs 15834 . . . . . . . 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 693 . . . . . . 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 2655 . . . . . 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 789 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵)
23 ovexd 6640 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) ∈ V)
24 simpr3 1067 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
25 eqid 2621 . . . . . . . 8 ((𝑊s 𝐴) ↾s 𝐵) = ((𝑊s 𝐴) ↾s 𝐵)
26 eqid 2621 . . . . . . . 8 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
2725, 26ressval2 15861 . . . . . . 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 1323 . . . . . 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 3816 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
30 sstr 3595 . . . . . . . . 9 (((Base‘𝑊) ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐴) → (Base‘𝑊) ⊆ 𝐴)
3129, 30mpan2 706 . . . . . . . 8 ((Base‘𝑊) ⊆ (𝐴𝐵) → (Base‘𝑊) ⊆ 𝐴)
321, 31nsyl 135 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘𝑊) ⊆ (𝐴𝐵))
33 inex1g 4766 . . . . . . . 8 (𝐴𝑋 → (𝐴𝐵) ∈ V)
343, 33syl 17 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴𝐵) ∈ V)
35 eqid 2621 . . . . . . . 8 (𝑊s (𝐴𝐵)) = (𝑊s (𝐴𝐵))
3635, 5ressval2 15861 . . . . . . 7 ((¬ (Base‘𝑊) ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
3732, 2, 34, 36syl3anc 1323 . . . . . 6 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
3821, 28, 373eqtr4d 2665 . . . . 5 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
3938exp31 629 . . . 4 (¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 → (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))))
40 ovex 6638 . . . . . . . 8 (𝑊s 𝐴) ∈ V
4125, 26ressid2 15860 . . . . . . . 8 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊s 𝐴) ∈ V ∧ 𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
4240, 41mp3an2 1409 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
43423ad2antr3 1226 . . . . . 6 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
44 in32 3808 . . . . . . . . 9 ((𝐴𝐵) ∩ (Base‘𝑊)) = ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵)
45 simpr2 1066 . . . . . . . . . . . 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 473 . . . . . . . . . . 11 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (Base‘(𝑊s 𝐴)) ⊆ 𝐵)
4846, 47eqsstrd 3623 . . . . . . . . . 10 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵)
49 df-ss 3573 . . . . . . . . . 10 ((𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵 ↔ ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
5048, 49sylib 208 . . . . . . . . 9 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
5144, 50syl5req 2668 . . . . . . . 8 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) = ((𝐴𝐵) ∩ (Base‘𝑊)))
5251oveq2d 6626 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴 ∩ (Base‘𝑊))) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
535ressinbas 15868 . . . . . . . 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 15868 . . . . . . . 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 2665 . . . . . 6 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
5843, 57eqtrd 2655 . . . . 5 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
5958ex 450 . . . 4 ((Base‘(𝑊s 𝐴)) ⊆ 𝐵 → ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
604, 5ressid2 15860 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑋) → (𝑊s 𝐴) = 𝑊)
61603adant3r3 1273 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = 𝑊)
6261oveq1d 6625 . . . . . 6 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐵))
63 inss2 3817 . . . . . . . . . . 11 (𝐵 ∩ (Base‘𝑊)) ⊆ (Base‘𝑊)
64 simpl 473 . . . . . . . . . . 11 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (Base‘𝑊) ⊆ 𝐴)
6563, 64syl5ss 3598 . . . . . . . . . 10 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴)
66 sseqin2 3800 . . . . . . . . . 10 ((𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
6765, 66sylib 208 . . . . . . . . 9 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
688, 67syl5req 2668 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘𝑊)) = ((𝐴𝐵) ∩ (Base‘𝑊)))
6968oveq2d 6626 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐵 ∩ (Base‘𝑊))) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
70 simpr3 1067 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
715ressinbas 15868 . . . . . . . 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 1066 . . . . . . . 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 2665 . . . . . 6 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐵) = (𝑊s (𝐴𝐵)))
7662, 75eqtrd 2655 . . . . 5 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
7776ex 450 . . . 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 1265 . 2 (𝑊 ∈ V → ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
80 ress0 15866 . . . 4 (∅ ↾s 𝐵) = ∅
81 reldmress 15858 . . . . . 6 Rel dom ↾s
8281ovprc1 6644 . . . . 5 𝑊 ∈ V → (𝑊s 𝐴) = ∅)
8382oveq1d 6625 . . . 4 𝑊 ∈ V → ((𝑊s 𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
8481ovprc1 6644 . . . 4 𝑊 ∈ V → (𝑊s (𝐴𝐵)) = ∅)
8580, 83, 843eqtr4a 2681 . . 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 1036   = wceq 1480  wcel 1987  Vcvv 3189  cin 3558  wss 3559  c0 3896  cop 4159  cfv 5852  (class class class)co 6610  ndxcnx 15789   sSet csts 15790  Basecbs 15792  s cress 15793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-i2m1 9956  ax-1ne0 9957  ax-rrecex 9960  ax-cnre 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-nn 10973  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799
This theorem is referenced by:  ressabs  15871  xrge00  29495  xrge0slmod  29653  esumpfinvallem  29941  lmhmlnmsplit  37172
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