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Theorem funcres2c 17852
Description: Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
Hypotheses
Ref Expression
funcres2c.a 𝐴 = (Baseβ€˜πΆ)
funcres2c.e 𝐸 = (𝐷 β†Ύs 𝑆)
funcres2c.d (πœ‘ β†’ 𝐷 ∈ Cat)
funcres2c.r (πœ‘ β†’ 𝑆 ∈ 𝑉)
funcres2c.1 (πœ‘ β†’ 𝐹:π΄βŸΆπ‘†)
Assertion
Ref Expression
funcres2c (πœ‘ β†’ (𝐹(𝐢 Func 𝐷)𝐺 ↔ 𝐹(𝐢 Func 𝐸)𝐺))

Proof of Theorem funcres2c
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 orc 866 . . 3 (𝐹(𝐢 Func 𝐷)𝐺 β†’ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺))
21a1i 11 . 2 (πœ‘ β†’ (𝐹(𝐢 Func 𝐷)𝐺 β†’ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)))
3 olc 867 . . 3 (𝐹(𝐢 Func 𝐸)𝐺 β†’ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺))
43a1i 11 . 2 (πœ‘ β†’ (𝐹(𝐢 Func 𝐸)𝐺 β†’ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)))
5 funcres2c.a . . . . 5 𝐴 = (Baseβ€˜πΆ)
6 eqid 2733 . . . . 5 (Hom β€˜πΆ) = (Hom β€˜πΆ)
7 eqid 2733 . . . . . . 7 (Baseβ€˜π·) = (Baseβ€˜π·)
8 eqid 2733 . . . . . . 7 (Homf β€˜π·) = (Homf β€˜π·)
9 funcres2c.d . . . . . . 7 (πœ‘ β†’ 𝐷 ∈ Cat)
10 inss2 4230 . . . . . . . 8 (𝑆 ∩ (Baseβ€˜π·)) βŠ† (Baseβ€˜π·)
1110a1i 11 . . . . . . 7 (πœ‘ β†’ (𝑆 ∩ (Baseβ€˜π·)) βŠ† (Baseβ€˜π·))
127, 8, 9, 11fullsubc 17800 . . . . . 6 (πœ‘ β†’ ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))) ∈ (Subcatβ€˜π·))
1312adantr 482 . . . . 5 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))) ∈ (Subcatβ€˜π·))
148, 7homffn 17637 . . . . . . 7 (Homf β€˜π·) Fn ((Baseβ€˜π·) Γ— (Baseβ€˜π·))
15 xpss12 5692 . . . . . . . 8 (((𝑆 ∩ (Baseβ€˜π·)) βŠ† (Baseβ€˜π·) ∧ (𝑆 ∩ (Baseβ€˜π·)) βŠ† (Baseβ€˜π·)) β†’ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))) βŠ† ((Baseβ€˜π·) Γ— (Baseβ€˜π·)))
1610, 10, 15mp2an 691 . . . . . . 7 ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))) βŠ† ((Baseβ€˜π·) Γ— (Baseβ€˜π·))
17 fnssres 6674 . . . . . . 7 (((Homf β€˜π·) Fn ((Baseβ€˜π·) Γ— (Baseβ€˜π·)) ∧ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))) βŠ† ((Baseβ€˜π·) Γ— (Baseβ€˜π·))) β†’ ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))) Fn ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))))
1814, 16, 17mp2an 691 . . . . . 6 ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))) Fn ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))
1918a1i 11 . . . . 5 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))) Fn ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))))
20 funcres2c.1 . . . . . . . 8 (πœ‘ β†’ 𝐹:π΄βŸΆπ‘†)
2120adantr 482 . . . . . . 7 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ 𝐹:π΄βŸΆπ‘†)
2221ffnd 6719 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ 𝐹 Fn 𝐴)
2321frnd 6726 . . . . . . 7 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ ran 𝐹 βŠ† 𝑆)
24 simpr 486 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐷)𝐺) β†’ 𝐹(𝐢 Func 𝐷)𝐺)
255, 7, 24funcf1 17816 . . . . . . . . 9 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐷)𝐺) β†’ 𝐹:𝐴⟢(Baseβ€˜π·))
2625frnd 6726 . . . . . . . 8 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐷)𝐺) β†’ ran 𝐹 βŠ† (Baseβ€˜π·))
27 eqid 2733 . . . . . . . . . . 11 (Baseβ€˜πΈ) = (Baseβ€˜πΈ)
28 simpr 486 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ 𝐹(𝐢 Func 𝐸)𝐺)
295, 27, 28funcf1 17816 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ 𝐹:𝐴⟢(Baseβ€˜πΈ))
3029frnd 6726 . . . . . . . . 9 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ ran 𝐹 βŠ† (Baseβ€˜πΈ))
31 funcres2c.e . . . . . . . . . 10 𝐸 = (𝐷 β†Ύs 𝑆)
3231, 7ressbasss 17183 . . . . . . . . 9 (Baseβ€˜πΈ) βŠ† (Baseβ€˜π·)
3330, 32sstrdi 3995 . . . . . . . 8 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ ran 𝐹 βŠ† (Baseβ€˜π·))
3426, 33jaodan 957 . . . . . . 7 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ ran 𝐹 βŠ† (Baseβ€˜π·))
3523, 34ssind 4233 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ ran 𝐹 βŠ† (𝑆 ∩ (Baseβ€˜π·)))
36 df-f 6548 . . . . . 6 (𝐹:𝐴⟢(𝑆 ∩ (Baseβ€˜π·)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† (𝑆 ∩ (Baseβ€˜π·))))
3722, 35, 36sylanbrc 584 . . . . 5 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ 𝐹:𝐴⟢(𝑆 ∩ (Baseβ€˜π·)))
38 eqid 2733 . . . . . . . . 9 (Hom β€˜π·) = (Hom β€˜π·)
39 simpr 486 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐷)𝐺) β†’ 𝐹(𝐢 Func 𝐷)𝐺)
40 simplrl 776 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐷)𝐺) β†’ π‘₯ ∈ 𝐴)
41 simplrr 777 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐷)𝐺) β†’ 𝑦 ∈ 𝐴)
425, 6, 38, 39, 40, 41funcf2 17818 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐷)𝐺) β†’ (π‘₯𝐺𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((πΉβ€˜π‘₯)(Hom β€˜π·)(πΉβ€˜π‘¦)))
43 eqid 2733 . . . . . . . . . 10 (Hom β€˜πΈ) = (Hom β€˜πΈ)
44 simpr 486 . . . . . . . . . 10 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ 𝐹(𝐢 Func 𝐸)𝐺)
45 simplrl 776 . . . . . . . . . 10 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ π‘₯ ∈ 𝐴)
46 simplrr 777 . . . . . . . . . 10 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ 𝑦 ∈ 𝐴)
475, 6, 43, 44, 45, 46funcf2 17818 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ (π‘₯𝐺𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((πΉβ€˜π‘₯)(Hom β€˜πΈ)(πΉβ€˜π‘¦)))
48 funcres2c.r . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑆 ∈ 𝑉)
4931, 38resshom 17364 . . . . . . . . . . . . 13 (𝑆 ∈ 𝑉 β†’ (Hom β€˜π·) = (Hom β€˜πΈ))
5048, 49syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ (Hom β€˜π·) = (Hom β€˜πΈ))
5150ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ (Hom β€˜π·) = (Hom β€˜πΈ))
5251oveqd 7426 . . . . . . . . . 10 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ ((πΉβ€˜π‘₯)(Hom β€˜π·)(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘₯)(Hom β€˜πΈ)(πΉβ€˜π‘¦)))
5352feq3d 6705 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ ((π‘₯𝐺𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((πΉβ€˜π‘₯)(Hom β€˜π·)(πΉβ€˜π‘¦)) ↔ (π‘₯𝐺𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((πΉβ€˜π‘₯)(Hom β€˜πΈ)(πΉβ€˜π‘¦))))
5447, 53mpbird 257 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ (π‘₯𝐺𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((πΉβ€˜π‘₯)(Hom β€˜π·)(πΉβ€˜π‘¦)))
5542, 54jaodan 957 . . . . . . 7 (((πœ‘ ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (π‘₯𝐺𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((πΉβ€˜π‘₯)(Hom β€˜π·)(πΉβ€˜π‘¦)))
5655an32s 651 . . . . . 6 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (π‘₯𝐺𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((πΉβ€˜π‘₯)(Hom β€˜π·)(πΉβ€˜π‘¦)))
5737adantr 482 . . . . . . . . . 10 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ 𝐹:𝐴⟢(𝑆 ∩ (Baseβ€˜π·)))
58 simprl 770 . . . . . . . . . 10 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ π‘₯ ∈ 𝐴)
5957, 58ffvelcdmd 7088 . . . . . . . . 9 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (πΉβ€˜π‘₯) ∈ (𝑆 ∩ (Baseβ€˜π·)))
60 simprr 772 . . . . . . . . . 10 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ 𝑦 ∈ 𝐴)
6157, 60ffvelcdmd 7088 . . . . . . . . 9 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (πΉβ€˜π‘¦) ∈ (𝑆 ∩ (Baseβ€˜π·)))
6259, 61ovresd 7574 . . . . . . . 8 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ ((πΉβ€˜π‘₯)((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))))(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘₯)(Homf β€˜π·)(πΉβ€˜π‘¦)))
6359elin2d 4200 . . . . . . . . 9 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (πΉβ€˜π‘₯) ∈ (Baseβ€˜π·))
6461elin2d 4200 . . . . . . . . 9 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (πΉβ€˜π‘¦) ∈ (Baseβ€˜π·))
658, 7, 38, 63, 64homfval 17636 . . . . . . . 8 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ ((πΉβ€˜π‘₯)(Homf β€˜π·)(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘₯)(Hom β€˜π·)(πΉβ€˜π‘¦)))
6662, 65eqtrd 2773 . . . . . . 7 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ ((πΉβ€˜π‘₯)((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))))(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘₯)(Hom β€˜π·)(πΉβ€˜π‘¦)))
6766feq3d 6705 . . . . . 6 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ ((π‘₯𝐺𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((πΉβ€˜π‘₯)((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))))(πΉβ€˜π‘¦)) ↔ (π‘₯𝐺𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((πΉβ€˜π‘₯)(Hom β€˜π·)(πΉβ€˜π‘¦))))
6856, 67mpbird 257 . . . . 5 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (π‘₯𝐺𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((πΉβ€˜π‘₯)((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))))(πΉβ€˜π‘¦)))
695, 6, 13, 19, 37, 68funcres2b 17847 . . . 4 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (𝐹(𝐢 Func 𝐷)𝐺 ↔ 𝐹(𝐢 Func (𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))))))𝐺))
70 eqidd 2734 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (Homf β€˜πΆ) = (Homf β€˜πΆ))
71 eqidd 2734 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (compfβ€˜πΆ) = (compfβ€˜πΆ))
727ressinbas 17190 . . . . . . . . . . 11 (𝑆 ∈ 𝑉 β†’ (𝐷 β†Ύs 𝑆) = (𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·))))
7348, 72syl 17 . . . . . . . . . 10 (πœ‘ β†’ (𝐷 β†Ύs 𝑆) = (𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·))))
7431, 73eqtrid 2785 . . . . . . . . 9 (πœ‘ β†’ 𝐸 = (𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·))))
7574fveq2d 6896 . . . . . . . 8 (πœ‘ β†’ (Homf β€˜πΈ) = (Homf β€˜(𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·)))))
76 eqid 2733 . . . . . . . . . 10 (𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·))) = (𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·)))
77 eqid 2733 . . . . . . . . . 10 (𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))))) = (𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))))
787, 8, 9, 11, 76, 77fullresc 17801 . . . . . . . . 9 (πœ‘ β†’ ((Homf β€˜(𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·)))) = (Homf β€˜(𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))))) ∧ (compfβ€˜(𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·)))) = (compfβ€˜(𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))))))))
7978simpld 496 . . . . . . . 8 (πœ‘ β†’ (Homf β€˜(𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·)))) = (Homf β€˜(𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))))))
8075, 79eqtrd 2773 . . . . . . 7 (πœ‘ β†’ (Homf β€˜πΈ) = (Homf β€˜(𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))))))
8180adantr 482 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (Homf β€˜πΈ) = (Homf β€˜(𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))))))
8274fveq2d 6896 . . . . . . . 8 (πœ‘ β†’ (compfβ€˜πΈ) = (compfβ€˜(𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·)))))
8378simprd 497 . . . . . . . 8 (πœ‘ β†’ (compfβ€˜(𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·)))) = (compfβ€˜(𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))))))
8482, 83eqtrd 2773 . . . . . . 7 (πœ‘ β†’ (compfβ€˜πΈ) = (compfβ€˜(𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))))))
8584adantr 482 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (compfβ€˜πΈ) = (compfβ€˜(𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))))))
86 df-br 5150 . . . . . . . . . . 11 (𝐹(𝐢 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐷))
87 funcrcl 17813 . . . . . . . . . . 11 (⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐷) β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
8886, 87sylbi 216 . . . . . . . . . 10 (𝐹(𝐢 Func 𝐷)𝐺 β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
8988simpld 496 . . . . . . . . 9 (𝐹(𝐢 Func 𝐷)𝐺 β†’ 𝐢 ∈ Cat)
90 df-br 5150 . . . . . . . . . . 11 (𝐹(𝐢 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐸))
91 funcrcl 17813 . . . . . . . . . . 11 (⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐸) β†’ (𝐢 ∈ Cat ∧ 𝐸 ∈ Cat))
9290, 91sylbi 216 . . . . . . . . . 10 (𝐹(𝐢 Func 𝐸)𝐺 β†’ (𝐢 ∈ Cat ∧ 𝐸 ∈ Cat))
9392simpld 496 . . . . . . . . 9 (𝐹(𝐢 Func 𝐸)𝐺 β†’ 𝐢 ∈ Cat)
9489, 93jaoi 856 . . . . . . . 8 ((𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺) β†’ 𝐢 ∈ Cat)
9594elexd 3495 . . . . . . 7 ((𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺) β†’ 𝐢 ∈ V)
9695adantl 483 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ 𝐢 ∈ V)
9731ovexi 7443 . . . . . . 7 𝐸 ∈ V
9897a1i 11 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ 𝐸 ∈ V)
99 ovexd 7444 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))))) ∈ V)
10070, 71, 81, 85, 96, 96, 98, 99funcpropd 17851 . . . . 5 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (𝐢 Func 𝐸) = (𝐢 Func (𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))))))
101100breqd 5160 . . . 4 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (𝐹(𝐢 Func 𝐸)𝐺 ↔ 𝐹(𝐢 Func (𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·))))))𝐺))
10269, 101bitr4d 282 . . 3 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (𝐹(𝐢 Func 𝐷)𝐺 ↔ 𝐹(𝐢 Func 𝐸)𝐺))
103102ex 414 . 2 (πœ‘ β†’ ((𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺) β†’ (𝐹(𝐢 Func 𝐷)𝐺 ↔ 𝐹(𝐢 Func 𝐸)𝐺)))
1042, 4, 103pm5.21ndd 381 1 (πœ‘ β†’ (𝐹(𝐢 Func 𝐷)𝐺 ↔ 𝐹(𝐢 Func 𝐸)𝐺))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948   βŠ† wss 3949  βŸ¨cop 4635   class class class wbr 5149   Γ— cxp 5675  ran crn 5678   β†Ύ cres 5679   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144   β†Ύs cress 17173  Hom chom 17208  Catccat 17608  Homf chomf 17610  compfccomf 17611   β†Ύcat cresc 17755  Subcatcsubc 17756   Func cfunc 17804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-hom 17221  df-cco 17222  df-cat 17612  df-cid 17613  df-homf 17614  df-comf 17615  df-ssc 17757  df-resc 17758  df-subc 17759  df-func 17808
This theorem is referenced by:  fthres2c  17882  fullres2c  17890
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