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Theorem funcres2c 17881
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 2727 . . . . 5 (Hom β€˜πΆ) = (Hom β€˜πΆ)
7 eqid 2727 . . . . . . 7 (Baseβ€˜π·) = (Baseβ€˜π·)
8 eqid 2727 . . . . . . 7 (Homf β€˜π·) = (Homf β€˜π·)
9 funcres2c.d . . . . . . 7 (πœ‘ β†’ 𝐷 ∈ Cat)
10 inss2 4225 . . . . . . . 8 (𝑆 ∩ (Baseβ€˜π·)) βŠ† (Baseβ€˜π·)
1110a1i 11 . . . . . . 7 (πœ‘ β†’ (𝑆 ∩ (Baseβ€˜π·)) βŠ† (Baseβ€˜π·))
127, 8, 9, 11fullsubc 17827 . . . . . 6 (πœ‘ β†’ ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))) ∈ (Subcatβ€˜π·))
1312adantr 480 . . . . 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 17664 . . . . . . 7 (Homf β€˜π·) Fn ((Baseβ€˜π·) Γ— (Baseβ€˜π·))
15 xpss12 5687 . . . . . . . 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 6672 . . . . . . 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 480 . . . . . . 7 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ 𝐹:π΄βŸΆπ‘†)
2221ffnd 6717 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ 𝐹 Fn 𝐴)
2321frnd 6724 . . . . . . 7 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ ran 𝐹 βŠ† 𝑆)
24 simpr 484 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐷)𝐺) β†’ 𝐹(𝐢 Func 𝐷)𝐺)
255, 7, 24funcf1 17843 . . . . . . . . 9 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐷)𝐺) β†’ 𝐹:𝐴⟢(Baseβ€˜π·))
2625frnd 6724 . . . . . . . 8 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐷)𝐺) β†’ ran 𝐹 βŠ† (Baseβ€˜π·))
27 eqid 2727 . . . . . . . . . . 11 (Baseβ€˜πΈ) = (Baseβ€˜πΈ)
28 simpr 484 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ 𝐹(𝐢 Func 𝐸)𝐺)
295, 27, 28funcf1 17843 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ 𝐹:𝐴⟢(Baseβ€˜πΈ))
3029frnd 6724 . . . . . . . . 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 17210 . . . . . . . . 9 (Baseβ€˜πΈ) βŠ† (Baseβ€˜π·)
3330, 32sstrdi 3990 . . . . . . . 8 ((πœ‘ ∧ 𝐹(𝐢 Func 𝐸)𝐺) β†’ ran 𝐹 βŠ† (Baseβ€˜π·))
3426, 33jaodan 956 . . . . . . 7 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ ran 𝐹 βŠ† (Baseβ€˜π·))
3523, 34ssind 4228 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ ran 𝐹 βŠ† (𝑆 ∩ (Baseβ€˜π·)))
36 df-f 6546 . . . . . 6 (𝐹:𝐴⟢(𝑆 ∩ (Baseβ€˜π·)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† (𝑆 ∩ (Baseβ€˜π·))))
3722, 35, 36sylanbrc 582 . . . . 5 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ 𝐹:𝐴⟢(𝑆 ∩ (Baseβ€˜π·)))
38 eqid 2727 . . . . . . . . 9 (Hom β€˜π·) = (Hom β€˜π·)
39 simpr 484 . . . . . . . . 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 17845 . . . . . . . 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 2727 . . . . . . . . . 10 (Hom β€˜πΈ) = (Hom β€˜πΈ)
44 simpr 484 . . . . . . . . . 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 17845 . . . . . . . . 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 17391 . . . . . . . . . . . . 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 7431 . . . . . . . . . 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 6703 . . . . . . . . 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 956 . . . . . . 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 480 . . . . . . . . . 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 7089 . . . . . . . . 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 7089 . . . . . . . . 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 7582 . . . . . . . 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 4195 . . . . . . . . 9 (((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) β†’ (πΉβ€˜π‘₯) ∈ (Baseβ€˜π·))
6461elin2d 4195 . . . . . . . . 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 17663 . . . . . . . 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 2767 . . . . . . 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 6703 . . . . . 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 17874 . . . 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 2728 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (Homf β€˜πΆ) = (Homf β€˜πΆ))
71 eqidd 2728 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ (compfβ€˜πΆ) = (compfβ€˜πΆ))
727ressinbas 17217 . . . . . . . . . . 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 2779 . . . . . . . . 9 (πœ‘ β†’ 𝐸 = (𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·))))
7574fveq2d 6895 . . . . . . . 8 (πœ‘ β†’ (Homf β€˜πΈ) = (Homf β€˜(𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·)))))
76 eqid 2727 . . . . . . . . . 10 (𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·))) = (𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·)))
77 eqid 2727 . . . . . . . . . 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 17828 . . . . . . . . 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 494 . . . . . . . 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 2767 . . . . . . 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 480 . . . . . 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 6895 . . . . . . . 8 (πœ‘ β†’ (compfβ€˜πΈ) = (compfβ€˜(𝐷 β†Ύs (𝑆 ∩ (Baseβ€˜π·)))))
8378simprd 495 . . . . . . . 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 2767 . . . . . . 7 (πœ‘ β†’ (compfβ€˜πΈ) = (compfβ€˜(𝐷 β†Ύcat ((Homf β€˜π·) β†Ύ ((𝑆 ∩ (Baseβ€˜π·)) Γ— (𝑆 ∩ (Baseβ€˜π·)))))))
8584adantr 480 . . . . . 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 5143 . . . . . . . . . . 11 (𝐹(𝐢 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐷))
87 funcrcl 17840 . . . . . . . . . . 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 494 . . . . . . . . 9 (𝐹(𝐢 Func 𝐷)𝐺 β†’ 𝐢 ∈ Cat)
90 df-br 5143 . . . . . . . . . . 11 (𝐹(𝐢 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐸))
91 funcrcl 17840 . . . . . . . . . . 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 494 . . . . . . . . 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 3490 . . . . . . 7 ((𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺) β†’ 𝐢 ∈ V)
9695adantl 481 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ 𝐢 ∈ V)
9731ovexi 7448 . . . . . . 7 𝐸 ∈ V
9897a1i 11 . . . . . 6 ((πœ‘ ∧ (𝐹(𝐢 Func 𝐷)𝐺 ∨ 𝐹(𝐢 Func 𝐸)𝐺)) β†’ 𝐸 ∈ V)
99 ovexd 7449 . . . . . 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 17880 . . . . 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 5153 . . . 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 412 . 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 379 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 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099  Vcvv 3469   ∩ cin 3943   βŠ† wss 3944  βŸ¨cop 4630   class class class wbr 5142   Γ— cxp 5670  ran crn 5673   β†Ύ cres 5674   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414  Basecbs 17171   β†Ύs cress 17200  Hom chom 17235  Catccat 17635  Homf chomf 17637  compfccomf 17638   β†Ύcat cresc 17782  Subcatcsubc 17783   Func cfunc 17831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8838  df-pm 8839  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-hom 17248  df-cco 17249  df-cat 17639  df-cid 17640  df-homf 17641  df-comf 17642  df-ssc 17784  df-resc 17785  df-subc 17786  df-func 17835
This theorem is referenced by:  fthres2c  17911  fullres2c  17919
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