Theorem imaid 50409

Description: An image of a functor preserves the identity morphism. (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)))
imassc.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
imaid.i	⊢ 𝐼 = (Id‘𝐸)
imaid.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)

Assertion

Ref	Expression
imaid	⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐾𝑋))

Distinct variable groups:   𝐹,𝑝,𝑥,𝑦   𝐺,𝑝,𝑥,𝑦   𝐻,𝑝,𝑥,𝑦   𝑥,𝑆,𝑦   𝐼,𝑝   𝑋,𝑝,𝑥,𝑦

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

Proof of Theorem imaid

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaid.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
2		imasubc.s	. . . . . . 7 ⊢ 𝑆 = (𝐹 “ 𝐴)
3	1, 2	eleqtrdi 2873	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐹 “ 𝐴))
4		inisegn0a 50086	. . . . . 6 ⊢ (𝑋 ∈ (𝐹 “ 𝐴) → (◡𝐹 “ {𝑋}) ≠ ∅)
5	3, 4	syl 18	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {𝑋}) ≠ ∅)
6		n0 4302	. . . . 5 ⊢ ((◡𝐹 “ {𝑋}) ≠ ∅ ↔ ∃𝑚 𝑚 ∈ (◡𝐹 “ {𝑋}))
7	5, 6	sylib 221	. . . 4 ⊢ (𝜑 → ∃𝑚 𝑚 ∈ (◡𝐹 “ {𝑋}))
8		fveq2 6893	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐺‘𝑝) = (𝐺‘⟨𝑚, 𝑚⟩))
9		df-ov 7430	. . . . . . . 8 ⊢ (𝑚𝐺𝑚) = (𝐺‘⟨𝑚, 𝑚⟩)
10	8, 9	eqtr4di 2816	. . . . . . 7 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐺‘𝑝) = (𝑚𝐺𝑚))
11		fveq2 6893	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐻‘𝑝) = (𝐻‘⟨𝑚, 𝑚⟩))
12		df-ov 7430	. . . . . . . 8 ⊢ (𝑚𝐻𝑚) = (𝐻‘⟨𝑚, 𝑚⟩)
13	11, 12	eqtr4di 2816	. . . . . . 7 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐻‘𝑝) = (𝑚𝐻𝑚))
14	10, 13	imaeq12d 6058	. . . . . 6 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → ((𝐺‘𝑝) “ (𝐻‘𝑝)) = ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
15	14	eleq2d 2849	. . . . 5 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → ((𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)) ↔ (𝐼‘𝑋) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚))))
16		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝑚 ∈ (◡𝐹 “ {𝑋}))
17	16, 16	opelxpd 5693	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ⟨𝑚, 𝑚⟩ ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋})))
18		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
19		eqid 2763	. . . . . . . 8 ⊢ (Id‘𝐷) = (Id‘𝐷)
20		imaid.i	. . . . . . . 8 ⊢ 𝐼 = (Id‘𝐸)
21		imassc.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
22	21	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝐹(𝐷 Func 𝐸)𝐺)
23		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘𝐸) = (Base‘𝐸)
24	18, 23, 21	funcf1 18053	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(Base‘𝐷)⟶(Base‘𝐸))
25	24	ffnd 6718	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐷))
26		fniniseg 7067	. . . . . . . . . . 11 ⊢ (𝐹 Fn (Base‘𝐷) → (𝑚 ∈ (◡𝐹 “ {𝑋}) ↔ (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋)))
27	25, 26	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ (◡𝐹 “ {𝑋}) ↔ (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋)))
28	27	biimpa 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋))
29	28	simpld 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝑚 ∈ (Base‘𝐷))
30	18, 19, 20, 22, 29	funcid 18057	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) = (𝐼‘(𝐹‘𝑚)))
31	28	simprd 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐹‘𝑚) = 𝑋)
32	31	fveq2d 6897	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐼‘(𝐹‘𝑚)) = (𝐼‘𝑋))
33	30, 32	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) = (𝐼‘𝑋))
34		imasubc.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐷)
35	22	funcrcl2 50332	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝐷 ∈ Cat)
36	18, 34, 19, 35, 29	catidcl 17868	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((Id‘𝐷)‘𝑚) ∈ (𝑚𝐻𝑚))
37		eqid 2763	. . . . . . . . 9 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
38	18, 34, 37, 22, 29, 29	funcf2 18055	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝑚𝐺𝑚):(𝑚𝐻𝑚)⟶((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑚)))
39	38	funfvima2d 7245	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) ∧ ((Id‘𝐷)‘𝑚) ∈ (𝑚𝐻𝑚)) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
40	36, 39	mpdan 700	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
41	33, 40	eqeltrrd 2864	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐼‘𝑋) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
42	15, 17, 41	rspcedvdw 3582	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ∃𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))(𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)))
43	7, 42	exlimddv 1969	. . 3 ⊢ (𝜑 → ∃𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))(𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)))
44	43	eliund 4960	. 2 ⊢ (𝜑 → (𝐼‘𝑋) ∈ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
45		relfunc 18049	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
46	45	brrelex1i 5710	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝐹 ∈ V)
47	21, 46	syl 18	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
48		imasubc.k	. . 3 ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
49	47, 47, 1, 1, 48	imasubclem3 50361	. 2 ⊢ (𝜑 → (𝑋𝐾𝑋) = ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
50	44, 49	eleqtrrd 2866	1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐾𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ≠ wne 2958  ∃wrex 3089  Vcvv 3453  ∅c0 4280  {csn 4587  ⟨cop 4837  ∪ ciun 4953   class class class wbr 5105   × cxp 5652  ^◡ccnv 5653   “ cima 5657   Fn wfn 6541  ‘cfv 6546  (class class class)co 7427   ∈ cmpo 7429  Basecbs 17399  Hom chom 17451  Idccid 17851   Func cfunc 18041

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-map 8860  df-ixp 8937  df-cat 17854  df-cid 17855  df-func 18045

This theorem is used by:  imasubc3  50411