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Theorem rescabsOLD 17785
Description: Obsolete proof of seqp1d 13985 as of 10-Nov-2024. Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
rescabs.c (πœ‘ β†’ 𝐢 ∈ 𝑉)
rescabs.h (πœ‘ β†’ 𝐻 Fn (𝑆 Γ— 𝑆))
rescabs.j (πœ‘ β†’ 𝐽 Fn (𝑇 Γ— 𝑇))
rescabs.s (πœ‘ β†’ 𝑆 ∈ π‘Š)
rescabs.t (πœ‘ β†’ 𝑇 βŠ† 𝑆)
Assertion
Ref Expression
rescabsOLD (πœ‘ β†’ ((𝐢 β†Ύcat 𝐻) β†Ύcat 𝐽) = (𝐢 β†Ύcat 𝐽))
Proof of Theorem rescabsOLD
StepHypRef Expression
1 eqid 2732 . . . 4 (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύcat 𝐽) = (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύcat 𝐽)
2 ovexd 7446 . . . 4 (πœ‘ β†’ ((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) ∈ V)
3 rescabs.s . . . . 5 (πœ‘ β†’ 𝑆 ∈ π‘Š)
4 rescabs.t . . . . 5 (πœ‘ β†’ 𝑇 βŠ† 𝑆)
53, 4ssexd 5324 . . . 4 (πœ‘ β†’ 𝑇 ∈ V)
6 rescabs.j . . . 4 (πœ‘ β†’ 𝐽 Fn (𝑇 Γ— 𝑇))
71, 2, 5, 6rescval2 17777 . . 3 (πœ‘ β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύcat 𝐽) = ((((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
8 simpr 485 . . . . . . 7 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇)
9 ovexd 7446 . . . . . . 7 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ ((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) ∈ V)
105adantr 481 . . . . . . 7 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ 𝑇 ∈ V)
11 eqid 2732 . . . . . . . 8 (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) = (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇)
12 baseid 17149 . . . . . . . . 9 Base = Slot (Baseβ€˜ndx)
13 1re 11216 . . . . . . . . . . 11 1 ∈ ℝ
14 1nn 12225 . . . . . . . . . . . 12 1 ∈ β„•
15 4nn0 12493 . . . . . . . . . . . 12 4 ∈ β„•0
16 1nn0 12490 . . . . . . . . . . . 12 1 ∈ β„•0
17 1lt10 12818 . . . . . . . . . . . 12 1 < 10
1814, 15, 16, 17declti 12717 . . . . . . . . . . 11 1 < 14
1913, 18ltneii 11329 . . . . . . . . . 10 1 β‰  14
20 basendx 17155 . . . . . . . . . . 11 (Baseβ€˜ndx) = 1
21 homndx 17358 . . . . . . . . . . 11 (Hom β€˜ndx) = 14
2220, 21neeq12i 3007 . . . . . . . . . 10 ((Baseβ€˜ndx) β‰  (Hom β€˜ndx) ↔ 1 β‰  14)
2319, 22mpbir 230 . . . . . . . . 9 (Baseβ€˜ndx) β‰  (Hom β€˜ndx)
2412, 23setsnid 17144 . . . . . . . 8 (Baseβ€˜(𝐢 β†Ύs 𝑆)) = (Baseβ€˜((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩))
2511, 24ressid2 17179 . . . . . . 7 (((Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇 ∧ ((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) ∈ V ∧ 𝑇 ∈ V) β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) = ((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩))
268, 9, 10, 25syl3anc 1371 . . . . . 6 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) = ((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩))
2726oveq1d 7426 . . . . 5 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ ((((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩) = (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
28 ovex 7444 . . . . . 6 (𝐢 β†Ύs 𝑆) ∈ V
295, 5xpexd 7740 . . . . . . . 8 (πœ‘ β†’ (𝑇 Γ— 𝑇) ∈ V)
30 fnex 7221 . . . . . . . 8 ((𝐽 Fn (𝑇 Γ— 𝑇) ∧ (𝑇 Γ— 𝑇) ∈ V) β†’ 𝐽 ∈ V)
316, 29, 30syl2anc 584 . . . . . . 7 (πœ‘ β†’ 𝐽 ∈ V)
3231adantr 481 . . . . . 6 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ 𝐽 ∈ V)
33 setsabs 17114 . . . . . 6 (((𝐢 β†Ύs 𝑆) ∈ V ∧ 𝐽 ∈ V) β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) sSet ⟨(Hom β€˜ndx), 𝐽⟩) = ((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
3428, 32, 33sylancr 587 . . . . 5 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) sSet ⟨(Hom β€˜ndx), 𝐽⟩) = ((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
35 eqid 2732 . . . . . . . . . . . . . 14 (𝐢 β†Ύs 𝑆) = (𝐢 β†Ύs 𝑆)
36 eqid 2732 . . . . . . . . . . . . . 14 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
3735, 36ressbas 17181 . . . . . . . . . . . . 13 (𝑆 ∈ π‘Š β†’ (𝑆 ∩ (Baseβ€˜πΆ)) = (Baseβ€˜(𝐢 β†Ύs 𝑆)))
383, 37syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑆 ∩ (Baseβ€˜πΆ)) = (Baseβ€˜(𝐢 β†Ύs 𝑆)))
3938sseq1d 4013 . . . . . . . . . . 11 (πœ‘ β†’ ((𝑆 ∩ (Baseβ€˜πΆ)) βŠ† 𝑇 ↔ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇))
4039biimpar 478 . . . . . . . . . 10 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (𝑆 ∩ (Baseβ€˜πΆ)) βŠ† 𝑇)
41 inss2 4229 . . . . . . . . . . 11 (𝑆 ∩ (Baseβ€˜πΆ)) βŠ† (Baseβ€˜πΆ)
4241a1i 11 . . . . . . . . . 10 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (𝑆 ∩ (Baseβ€˜πΆ)) βŠ† (Baseβ€˜πΆ))
4340, 42ssind 4232 . . . . . . . . 9 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (𝑆 ∩ (Baseβ€˜πΆ)) βŠ† (𝑇 ∩ (Baseβ€˜πΆ)))
444adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ 𝑇 βŠ† 𝑆)
4544ssrind 4235 . . . . . . . . 9 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (𝑇 ∩ (Baseβ€˜πΆ)) βŠ† (𝑆 ∩ (Baseβ€˜πΆ)))
4643, 45eqssd 3999 . . . . . . . 8 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (𝑆 ∩ (Baseβ€˜πΆ)) = (𝑇 ∩ (Baseβ€˜πΆ)))
4746oveq2d 7427 . . . . . . 7 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (𝐢 β†Ύs (𝑆 ∩ (Baseβ€˜πΆ))) = (𝐢 β†Ύs (𝑇 ∩ (Baseβ€˜πΆ))))
483adantr 481 . . . . . . . 8 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ 𝑆 ∈ π‘Š)
4936ressinbas 17192 . . . . . . . 8 (𝑆 ∈ π‘Š β†’ (𝐢 β†Ύs 𝑆) = (𝐢 β†Ύs (𝑆 ∩ (Baseβ€˜πΆ))))
5048, 49syl 17 . . . . . . 7 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (𝐢 β†Ύs 𝑆) = (𝐢 β†Ύs (𝑆 ∩ (Baseβ€˜πΆ))))
5136ressinbas 17192 . . . . . . . 8 (𝑇 ∈ V β†’ (𝐢 β†Ύs 𝑇) = (𝐢 β†Ύs (𝑇 ∩ (Baseβ€˜πΆ))))
5210, 51syl 17 . . . . . . 7 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (𝐢 β†Ύs 𝑇) = (𝐢 β†Ύs (𝑇 ∩ (Baseβ€˜πΆ))))
5347, 50, 523eqtr4d 2782 . . . . . 6 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (𝐢 β†Ύs 𝑆) = (𝐢 β†Ύs 𝑇))
5453oveq1d 7426 . . . . 5 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ ((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐽⟩) = ((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
5527, 34, 543eqtrd 2776 . . . 4 ((πœ‘ ∧ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ ((((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩) = ((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
56 simpr 485 . . . . . . . 8 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇)
57 ovexd 7446 . . . . . . . 8 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ ((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) ∈ V)
585adantr 481 . . . . . . . 8 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ 𝑇 ∈ V)
5911, 24ressval2 17180 . . . . . . . 8 ((Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇 ∧ ((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) ∈ V ∧ 𝑇 ∈ V) β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) = (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) sSet ⟨(Baseβ€˜ndx), (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆)))⟩))
6056, 57, 58, 59syl3anc 1371 . . . . . . 7 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) = (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) sSet ⟨(Baseβ€˜ndx), (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆)))⟩))
61 ovexd 7446 . . . . . . . 8 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (𝐢 β†Ύs 𝑆) ∈ V)
6223necomi 2995 . . . . . . . . 9 (Hom β€˜ndx) β‰  (Baseβ€˜ndx)
6362a1i 11 . . . . . . . 8 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (Hom β€˜ndx) β‰  (Baseβ€˜ndx))
64 rescabs.h . . . . . . . . . 10 (πœ‘ β†’ 𝐻 Fn (𝑆 Γ— 𝑆))
653, 3xpexd 7740 . . . . . . . . . 10 (πœ‘ β†’ (𝑆 Γ— 𝑆) ∈ V)
66 fnex 7221 . . . . . . . . . 10 ((𝐻 Fn (𝑆 Γ— 𝑆) ∧ (𝑆 Γ— 𝑆) ∈ V) β†’ 𝐻 ∈ V)
6764, 65, 66syl2anc 584 . . . . . . . . 9 (πœ‘ β†’ 𝐻 ∈ V)
6867adantr 481 . . . . . . . 8 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ 𝐻 ∈ V)
69 fvex 6904 . . . . . . . . . 10 (Baseβ€˜(𝐢 β†Ύs 𝑆)) ∈ V
7069inex2 5318 . . . . . . . . 9 (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆))) ∈ V
7170a1i 11 . . . . . . . 8 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆))) ∈ V)
72 fvex 6904 . . . . . . . . 9 (Hom β€˜ndx) ∈ V
73 fvex 6904 . . . . . . . . 9 (Baseβ€˜ndx) ∈ V
7472, 73setscom 17115 . . . . . . . 8 ((((𝐢 β†Ύs 𝑆) ∈ V ∧ (Hom β€˜ndx) β‰  (Baseβ€˜ndx)) ∧ (𝐻 ∈ V ∧ (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆))) ∈ V)) β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) sSet ⟨(Baseβ€˜ndx), (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆)))⟩) = (((𝐢 β†Ύs 𝑆) sSet ⟨(Baseβ€˜ndx), (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆)))⟩) sSet ⟨(Hom β€˜ndx), 𝐻⟩))
7561, 63, 68, 71, 74syl22anc 837 . . . . . . 7 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) sSet ⟨(Baseβ€˜ndx), (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆)))⟩) = (((𝐢 β†Ύs 𝑆) sSet ⟨(Baseβ€˜ndx), (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆)))⟩) sSet ⟨(Hom β€˜ndx), 𝐻⟩))
76 eqid 2732 . . . . . . . . . . 11 ((𝐢 β†Ύs 𝑆) β†Ύs 𝑇) = ((𝐢 β†Ύs 𝑆) β†Ύs 𝑇)
77 eqid 2732 . . . . . . . . . . 11 (Baseβ€˜(𝐢 β†Ύs 𝑆)) = (Baseβ€˜(𝐢 β†Ύs 𝑆))
7876, 77ressval2 17180 . . . . . . . . . 10 ((Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇 ∧ (𝐢 β†Ύs 𝑆) ∈ V ∧ 𝑇 ∈ V) β†’ ((𝐢 β†Ύs 𝑆) β†Ύs 𝑇) = ((𝐢 β†Ύs 𝑆) sSet ⟨(Baseβ€˜ndx), (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆)))⟩))
7956, 61, 58, 78syl3anc 1371 . . . . . . . . 9 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ ((𝐢 β†Ύs 𝑆) β†Ύs 𝑇) = ((𝐢 β†Ύs 𝑆) sSet ⟨(Baseβ€˜ndx), (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆)))⟩))
803adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ 𝑆 ∈ π‘Š)
814adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ 𝑇 βŠ† 𝑆)
82 ressabs 17196 . . . . . . . . . 10 ((𝑆 ∈ π‘Š ∧ 𝑇 βŠ† 𝑆) β†’ ((𝐢 β†Ύs 𝑆) β†Ύs 𝑇) = (𝐢 β†Ύs 𝑇))
8380, 81, 82syl2anc 584 . . . . . . . . 9 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ ((𝐢 β†Ύs 𝑆) β†Ύs 𝑇) = (𝐢 β†Ύs 𝑇))
8479, 83eqtr3d 2774 . . . . . . . 8 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ ((𝐢 β†Ύs 𝑆) sSet ⟨(Baseβ€˜ndx), (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆)))⟩) = (𝐢 β†Ύs 𝑇))
8584oveq1d 7426 . . . . . . 7 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Baseβ€˜ndx), (𝑇 ∩ (Baseβ€˜(𝐢 β†Ύs 𝑆)))⟩) sSet ⟨(Hom β€˜ndx), 𝐻⟩) = ((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐻⟩))
8660, 75, 853eqtrd 2776 . . . . . 6 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) = ((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐻⟩))
8786oveq1d 7426 . . . . 5 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ ((((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩) = (((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐻⟩) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
88 ovex 7444 . . . . . 6 (𝐢 β†Ύs 𝑇) ∈ V
8931adantr 481 . . . . . 6 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ 𝐽 ∈ V)
90 setsabs 17114 . . . . . 6 (((𝐢 β†Ύs 𝑇) ∈ V ∧ 𝐽 ∈ V) β†’ (((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐻⟩) sSet ⟨(Hom β€˜ndx), 𝐽⟩) = ((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
9188, 89, 90sylancr 587 . . . . 5 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ (((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐻⟩) sSet ⟨(Hom β€˜ndx), 𝐽⟩) = ((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
9287, 91eqtrd 2772 . . . 4 ((πœ‘ ∧ Β¬ (Baseβ€˜(𝐢 β†Ύs 𝑆)) βŠ† 𝑇) β†’ ((((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩) = ((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
9355, 92pm2.61dan 811 . . 3 (πœ‘ β†’ ((((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩) = ((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
947, 93eqtrd 2772 . 2 (πœ‘ β†’ (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύcat 𝐽) = ((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
95 eqid 2732 . . . 4 (𝐢 β†Ύcat 𝐻) = (𝐢 β†Ύcat 𝐻)
96 rescabs.c . . . 4 (πœ‘ β†’ 𝐢 ∈ 𝑉)
9795, 96, 3, 64rescval2 17777 . . 3 (πœ‘ β†’ (𝐢 β†Ύcat 𝐻) = ((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩))
9897oveq1d 7426 . 2 (πœ‘ β†’ ((𝐢 β†Ύcat 𝐻) β†Ύcat 𝐽) = (((𝐢 β†Ύs 𝑆) sSet ⟨(Hom β€˜ndx), 𝐻⟩) β†Ύcat 𝐽))
99 eqid 2732 . . 3 (𝐢 β†Ύcat 𝐽) = (𝐢 β†Ύcat 𝐽)
10099, 96, 5, 6rescval2 17777 . 2 (πœ‘ β†’ (𝐢 β†Ύcat 𝐽) = ((𝐢 β†Ύs 𝑇) sSet ⟨(Hom β€˜ndx), 𝐽⟩))
10194, 98, 1003eqtr4d 2782 1 (πœ‘ β†’ ((𝐢 β†Ύcat 𝐻) β†Ύcat 𝐽) = (𝐢 β†Ύcat 𝐽))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  βŸ¨cop 4634   Γ— cxp 5674   Fn wfn 6538  β€˜cfv 6543  (class class class)co 7411  1c1 11113  4c4 12271  cdc 12679   sSet csts 17098  ndxcnx 17128  Basecbs 17146   β†Ύs cress 17175  Hom chom 17210   β†Ύcat cresc 17757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12475  df-z 12561  df-dec 12680  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-hom 17223  df-resc 17760
This theorem is referenced by: (None)
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