Theorem imasubc2 50182

Description: An image of a full functor is a (full) subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.)

Hypotheses

Ref	Expression
imasubc.s	⊢ 𝑆 = (𝐹 “ 𝐴)
imasubc.h	⊢ 𝐻 = (Hom ‘𝐷)
imasubc.k	⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
imasubc.f	⊢ (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺)

Assertion

Ref	Expression
imasubc2	⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))

Distinct variable groups:   𝐹,𝑝,𝑥,𝑦   𝐺,𝑝,𝑥,𝑦   𝐻,𝑝,𝑥,𝑦   𝑥,𝑆,𝑦   𝐸,𝑝   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑝)   𝐴(𝑥, 𝑦, 𝑝)   𝐷(𝑥, 𝑦, 𝑝)   𝑆(𝑝)   𝐸(𝑥, 𝑦)   𝐾(𝑥, 𝑦, 𝑝)

Proof of Theorem imasubc2

Step	Hyp	Ref	Expression
1		imasubc.s	. . . 4 ⊢ 𝑆 = (𝐹 “ 𝐴)
2		imasubc.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
3		imasubc.k	. . . 4 ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
4		imasubc.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺)
5		eqid 2760	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
6		eqid 2760	. . . 4 ⊢ (Homf ‘𝐸) = (Homf ‘𝐸)
7	1, 2, 3, 4, 5, 6	imasubc 50181	. . 3 ⊢ (𝜑 → (𝐾 Fn (𝑆 × 𝑆) ∧ 𝑆 ⊆ (Base‘𝐸) ∧ ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) = 𝐾))
8	7	simp3d 1162	. 2 ⊢ (𝜑 → ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) = 𝐾)
9		fullfunc 18044	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
10	9	ssbri 5149	. . . . 5 ⊢ (𝐹(𝐷 Full 𝐸)𝐺 → 𝐹(𝐷 Func 𝐸)𝐺)
11	4, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
12	11	funcrcl3 50110	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
13	7	simp2d 1161	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐸))
14	5, 6, 12, 13	fullsubc 17986	. 2 ⊢ (𝜑 → ((Homf ‘𝐸) ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐸))
15	8, 14	eqeltrrd 2861	1 ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ⊆ wss 3898  {csn 4583  ∪ ciun 4950   class class class wbr 5102   × cxp 5645  ^◡ccnv 5646   ↾ cres 5649   “ cima 5650   Fn wfn 6522  ‘cfv 6527  (class class class)co 7408   ∈ cmpo 7410  Basecbs 17348  Hom chom 17400  Hom_f chomf 17801  Subcatcsubc 17945   Func cfunc 17990   Full cful 18040

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-map 8827  df-pm 8828  df-ixp 8904  df-cat 17803  df-cid 17804  df-homf 17805  df-ssc 17946  df-subc 17948  df-func 17994  df-full 18042

This theorem is used by: (None)