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Theorem ressress 16678
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 769 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘𝑊) ⊆ 𝐴)
2 simpr1 1195 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝑊 ∈ V)
3 simpr2 1196 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐴𝑋)
4 eqid 2739 . . . . . . . . . 10 (𝑊s 𝐴) = (𝑊s 𝐴)
5 eqid 2739 . . . . . . . . . 10 (Base‘𝑊) = (Base‘𝑊)
64, 5ressval2 16669 . . . . . . . . 9 ((¬ (Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑋) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
71, 2, 3, 6syl3anc 1372 . . . . . . . 8 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
8 inass 4120 . . . . . . . . . . 11 ((𝐴𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
9 in12 4121 . . . . . . . . . . 11 (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
108, 9eqtri 2762 . . . . . . . . . 10 ((𝐴𝐵) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
114, 5ressbas 16670 . . . . . . . . . . . 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 4113 . . . . . . . . . 10 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘(𝑊s 𝐴))))
1410, 13eqtr2id 2787 . . . . . . . . 9 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘(𝑊s 𝐴))) = ((𝐴𝐵) ∩ (Base‘𝑊)))
1514opeq2d 4778 . . . . . . . 8 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊s 𝐴)))⟩ = ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩)
167, 15oveq12d 7201 . . . . . . 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 6700 . . . . . . . . 9 (Base‘𝑊) ∈ V
1817inex2 5196 . . . . . . . 8 ((𝐴𝐵) ∩ (Base‘𝑊)) ∈ V
19 setsabs 16642 . . . . . . . 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 589 . . . . . . 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 2774 . . . . . 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 767 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵)
23 ovexd 7218 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) ∈ V)
24 simpr3 1197 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
25 eqid 2739 . . . . . . . 8 ((𝑊s 𝐴) ↾s 𝐵) = ((𝑊s 𝐴) ↾s 𝐵)
26 eqid 2739 . . . . . . . 8 (Base‘(𝑊s 𝐴)) = (Base‘(𝑊s 𝐴))
2725, 26ressval2 16669 . . . . . . 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 1372 . . . . . 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 4129 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
30 sstr 3895 . . . . . . . . 9 (((Base‘𝑊) ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐴) → (Base‘𝑊) ⊆ 𝐴)
3129, 30mpan2 691 . . . . . . . 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 5197 . . . . . . . 8 (𝐴𝑋 → (𝐴𝐵) ∈ V)
343, 33syl 17 . . . . . . 7 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴𝐵) ∈ V)
35 eqid 2739 . . . . . . . 8 (𝑊s (𝐴𝐵)) = (𝑊s (𝐴𝐵))
3635, 5ressval2 16669 . . . . . . 7 ((¬ (Base‘𝑊) ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
3732, 2, 34, 36syl3anc 1372 . . . . . 6 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ (Base‘𝑊))⟩))
3821, 28, 373eqtr4d 2784 . . . . 5 (((¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
3938exp31 423 . . . 4 (¬ (Base‘(𝑊s 𝐴)) ⊆ 𝐵 → (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))))
40 ovex 7216 . . . . . . . 8 (𝑊s 𝐴) ∈ V
4125, 26ressid2 16668 . . . . . . . 8 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊s 𝐴) ∈ V ∧ 𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
4240, 41mp3an2 1450 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
43423ad2antr3 1191 . . . . . 6 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐴))
44 in32 4122 . . . . . . . . 9 ((𝐴𝐵) ∩ (Base‘𝑊)) = ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵)
45 simpr2 1196 . . . . . . . . . . . 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 486 . . . . . . . . . . 11 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (Base‘(𝑊s 𝐴)) ⊆ 𝐵)
4846, 47eqsstrd 3925 . . . . . . . . . 10 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵)
49 df-ss 3870 . . . . . . . . . 10 ((𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵 ↔ ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
5048, 49sylib 221 . . . . . . . . 9 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
5144, 50eqtr2id 2787 . . . . . . . 8 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (Base‘𝑊)) = ((𝐴𝐵) ∩ (Base‘𝑊)))
5251oveq2d 7199 . . . . . . 7 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐴 ∩ (Base‘𝑊))) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
535ressinbas 16676 . . . . . . . 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 16676 . . . . . . . 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 2784 . . . . . 6 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
5843, 57eqtrd 2774 . . . . 5 (((Base‘(𝑊s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
5958ex 416 . . . 4 ((Base‘(𝑊s 𝐴)) ⊆ 𝐵 → ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
604, 5ressid2 16668 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑋) → (𝑊s 𝐴) = 𝑊)
61603adant3r3 1185 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐴) = 𝑊)
6261oveq1d 7198 . . . . . 6 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐵))
63 inss2 4130 . . . . . . . . . . 11 (𝐵 ∩ (Base‘𝑊)) ⊆ (Base‘𝑊)
64 simpl 486 . . . . . . . . . . 11 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (Base‘𝑊) ⊆ 𝐴)
6563, 64sstrid 3898 . . . . . . . . . 10 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴)
66 sseqin2 4116 . . . . . . . . . 10 ((𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
6765, 66sylib 221 . . . . . . . . 9 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
688, 67eqtr2id 2787 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝐵 ∩ (Base‘𝑊)) = ((𝐴𝐵) ∩ (Base‘𝑊)))
6968oveq2d 7199 . . . . . . 7 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s (𝐵 ∩ (Base‘𝑊))) = (𝑊s ((𝐴𝐵) ∩ (Base‘𝑊))))
70 simpr3 1197 . . . . . . . 8 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
715ressinbas 16676 . . . . . . . 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 1196 . . . . . . . 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 2784 . . . . . 6 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → (𝑊s 𝐵) = (𝑊s (𝐴𝐵)))
7662, 75eqtrd 2774 . . . . 5 (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌)) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
7776ex 416 . . . 4 ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
7839, 59, 77pm2.61ii 186 . . 3 ((𝑊 ∈ V ∧ 𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
79783expib 1123 . 2 (𝑊 ∈ V → ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
80 ress0 16674 . . . 4 (∅ ↾s 𝐵) = ∅
81 reldmress 16666 . . . . . 6 Rel dom ↾s
8281ovprc1 7222 . . . . 5 𝑊 ∈ V → (𝑊s 𝐴) = ∅)
8382oveq1d 7198 . . . 4 𝑊 ∈ V → ((𝑊s 𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
8481ovprc1 7222 . . . 4 𝑊 ∈ V → (𝑊s (𝐴𝐵)) = ∅)
8580, 83, 843eqtr4a 2800 . . 3 𝑊 ∈ V → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
8685a1d 25 . 2 𝑊 ∈ V → ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵))))
8779, 86pm2.61i 185 1 ((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  Vcvv 3400  cin 3852  wss 3853  c0 4221  cop 4532  cfv 6350  (class class class)co 7183  ndxcnx 16596   sSet csts 16597  Basecbs 16599  s cress 16600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492  ax-cnex 10684  ax-1cn 10686  ax-addcl 10688
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6186  df-on 6187  df-lim 6188  df-suc 6189  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7186  df-oprab 7187  df-mpo 7188  df-om 7613  df-wrecs 7989  df-recs 8050  df-rdg 8088  df-nn 11730  df-ndx 16602  df-slot 16603  df-base 16605  df-sets 16606  df-ress 16607
This theorem is referenced by:  ressabs  16679  xrge00  30885  xrge0slmod  31133  fldexttr  31318  esumpfinvallem  31625  lmhmlnmsplit  40525
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