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Theorem ressressg 12537
Description: Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
ressressg ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ ((π‘Š β†Ύs 𝐴) β†Ύs 𝐡) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
Proof of Theorem ressressg
StepHypRef Expression
1 eqidd 2178 . . . . . . 7 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs 𝐴))
2 eqidd 2178 . . . . . . 7 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š))
3 simp3 999 . . . . . . 7 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ π‘Š ∈ 𝑍)
4 simp1 997 . . . . . . 7 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ 𝐴 ∈ 𝑋)
51, 2, 3, 4ressbasd 12530 . . . . . 6 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) = (Baseβ€˜(π‘Š β†Ύs 𝐴)))
65ineq2d 3338 . . . . 5 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (𝐡 ∩ (𝐴 ∩ (Baseβ€˜π‘Š))) = (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴))))
7 inass 3347 . . . . . 6 ((𝐡 ∩ 𝐴) ∩ (Baseβ€˜π‘Š)) = (𝐡 ∩ (𝐴 ∩ (Baseβ€˜π‘Š)))
8 incom 3329 . . . . . . 7 (𝐡 ∩ 𝐴) = (𝐴 ∩ 𝐡)
98ineq1i 3334 . . . . . 6 ((𝐡 ∩ 𝐴) ∩ (Baseβ€˜π‘Š)) = ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š))
107, 9eqtr3i 2200 . . . . 5 (𝐡 ∩ (𝐴 ∩ (Baseβ€˜π‘Š))) = ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š))
116, 10eqtr3di 2225 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴))) = ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š)))
1211opeq2d 3787 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴)))⟩ = ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š))⟩)
1312oveq2d 5894 . 2 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴)))⟩) = (π‘Š sSet ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š))⟩))
14 ressex 12528 . . . . 5 ((π‘Š ∈ 𝑍 ∧ 𝐴 ∈ 𝑋) β†’ (π‘Š β†Ύs 𝐴) ∈ V)
153, 4, 14syl2anc 411 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (π‘Š β†Ύs 𝐴) ∈ V)
16 simp2 998 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ 𝐡 ∈ π‘Œ)
17 ressvalsets 12527 . . . 4 (((π‘Š β†Ύs 𝐴) ∈ V ∧ 𝐡 ∈ π‘Œ) β†’ ((π‘Š β†Ύs 𝐴) β†Ύs 𝐡) = ((π‘Š β†Ύs 𝐴) sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴)))⟩))
1815, 16, 17syl2anc 411 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ ((π‘Š β†Ύs 𝐴) β†Ύs 𝐡) = ((π‘Š β†Ύs 𝐴) sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴)))⟩))
19 ressvalsets 12527 . . . . 5 ((π‘Š ∈ 𝑍 ∧ 𝐴 ∈ 𝑋) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
203, 4, 19syl2anc 411 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩))
2120oveq1d 5893 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ ((π‘Š β†Ύs 𝐴) sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴)))⟩) = ((π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩) sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴)))⟩))
22 basendxnn 12521 . . . . 5 (Baseβ€˜ndx) ∈ β„•
2322a1i 9 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (Baseβ€˜ndx) ∈ β„•)
24 inex1g 4141 . . . . 5 (𝐴 ∈ 𝑋 β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) ∈ V)
254, 24syl 14 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (𝐴 ∩ (Baseβ€˜π‘Š)) ∈ V)
26 inex1g 4141 . . . . 5 (𝐡 ∈ π‘Œ β†’ (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴))) ∈ V)
2716, 26syl 14 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴))) ∈ V)
283, 23, 25, 27setsabsd 12504 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ ((π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘Š))⟩) sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴)))⟩) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴)))⟩))
2918, 21, 283eqtrd 2214 . 2 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ ((π‘Š β†Ύs 𝐴) β†Ύs 𝐡) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜(π‘Š β†Ύs 𝐴)))⟩))
30 inex1g 4141 . . . 4 (𝐴 ∈ 𝑋 β†’ (𝐴 ∩ 𝐡) ∈ V)
314, 30syl 14 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (𝐴 ∩ 𝐡) ∈ V)
32 ressvalsets 12527 . . 3 ((π‘Š ∈ 𝑍 ∧ (𝐴 ∩ 𝐡) ∈ V) β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = (π‘Š sSet ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š))⟩))
333, 31, 32syl2anc 411 . 2 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = (π‘Š sSet ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ (Baseβ€˜π‘Š))⟩))
3413, 29, 333eqtr4d 2220 1 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ ∧ π‘Š ∈ 𝑍) β†’ ((π‘Š β†Ύs 𝐴) β†Ύs 𝐡) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ∩ cin 3130  βŸ¨cop 3597  β€˜cfv 5218  (class class class)co 5878  β„•cn 8922  ndxcnx 12462   sSet csts 12463  Basecbs 12465   β†Ύs cress 12466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-inn 8923  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-iress 12473
This theorem is referenced by:  ressabsg  12538
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