Theorem imaid 49775

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 2872	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐹 “ 𝐴))
4		inisegn0a 49457	. . . . . 6 ⊢ (𝑋 ∈ (𝐹 “ 𝐴) → (◡𝐹 “ {𝑋}) ≠ ∅)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {𝑋}) ≠ ∅)
6		n0 4305	. . . . 5 ⊢ ((◡𝐹 “ {𝑋}) ≠ ∅ ↔ ∃𝑚 𝑚 ∈ (◡𝐹 “ {𝑋}))
7	5, 6	sylib 220	. . . 4 ⊢ (𝜑 → ∃𝑚 𝑚 ∈ (◡𝐹 “ {𝑋}))
8		fveq2 6867	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐺‘𝑝) = (𝐺‘⟨𝑚, 𝑚⟩))
9		df-ov 7399	. . . . . . . 8 ⊢ (𝑚𝐺𝑚) = (𝐺‘⟨𝑚, 𝑚⟩)
10	8, 9	eqtr4di 2815	. . . . . . 7 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐺‘𝑝) = (𝑚𝐺𝑚))
11		fveq2 6867	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐻‘𝑝) = (𝐻‘⟨𝑚, 𝑚⟩))
12		df-ov 7399	. . . . . . . 8 ⊢ (𝑚𝐻𝑚) = (𝐻‘⟨𝑚, 𝑚⟩)
13	11, 12	eqtr4di 2815	. . . . . . 7 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → (𝐻‘𝑝) = (𝑚𝐻𝑚))
14	10, 13	imaeq12d 6050	. . . . . 6 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → ((𝐺‘𝑝) “ (𝐻‘𝑝)) = ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
15	14	eleq2d 2848	. . . . 5 ⊢ (𝑝 = ⟨𝑚, 𝑚⟩ → ((𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)) ↔ (𝐼‘𝑋) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚))))
16		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝑚 ∈ (◡𝐹 “ {𝑋}))
17	16, 16	opelxpd 5686	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ⟨𝑚, 𝑚⟩ ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋})))
18		eqid 2762	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
19		eqid 2762	. . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝐹(𝐷 Func 𝐸)𝐺)
23		eqid 2762	. . . . . . . . . . . . 13 ⊢ (Base‘𝐸) = (Base‘𝐸)
24	18, 23, 21	funcf1 17899	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(Base‘𝐷)⟶(Base‘𝐸))
25	24	ffnd 6692	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐷))
26		fniniseg 7041	. . . . . . . . . . 11 ⊢ (𝐹 Fn (Base‘𝐷) → (𝑚 ∈ (◡𝐹 “ {𝑋}) ↔ (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋)))
27	25, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ (◡𝐹 “ {𝑋}) ↔ (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋)))
28	27	biimpa 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝑚 ∈ (Base‘𝐷) ∧ (𝐹‘𝑚) = 𝑋))
29	28	simpld 498	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝑚 ∈ (Base‘𝐷))
30	18, 19, 20, 22, 29	funcid 17903	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) = (𝐼‘(𝐹‘𝑚)))
31	28	simprd 499	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐹‘𝑚) = 𝑋)
32	31	fveq2d 6871	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐼‘(𝐹‘𝑚)) = (𝐼‘𝑋))
33	30, 32	eqtrd 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) = (𝐼‘𝑋))
34		imasubc.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐷)
35	22	funcrcl2 49700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → 𝐷 ∈ Cat)
36	18, 34, 19, 35, 29	catidcl 17714	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((Id‘𝐷)‘𝑚) ∈ (𝑚𝐻𝑚))
37		eqid 2762	. . . . . . . . 9 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
38	18, 34, 37, 22, 29, 29	funcf2 17901	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝑚𝐺𝑚):(𝑚𝐻𝑚)⟶((𝐹‘𝑚)(Hom ‘𝐸)(𝐹‘𝑚)))
39	38	funfvima2d 7216	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) ∧ ((Id‘𝐷)‘𝑚) ∈ (𝑚𝐻𝑚)) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
40	36, 39	mpdan 697	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ((𝑚𝐺𝑚)‘((Id‘𝐷)‘𝑚)) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
41	33, 40	eqeltrrd 2863	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → (𝐼‘𝑋) ∈ ((𝑚𝐺𝑚) “ (𝑚𝐻𝑚)))
42	15, 17, 41	rspcedvdw 3584	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (◡𝐹 “ {𝑋})) → ∃𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))(𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)))
43	7, 42	exlimddv 1955	. . 3 ⊢ (𝜑 → ∃𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))(𝐼‘𝑋) ∈ ((𝐺‘𝑝) “ (𝐻‘𝑝)))
44	43	eliund 4956	. 2 ⊢ (𝜑 → (𝐼‘𝑋) ∈ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
45		relfunc 17895	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
46	45	brrelex1i 5703	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝐹 ∈ V)
47	21, 46	syl 17	. . 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 49727	. 2 ⊢ (𝜑 → (𝑋𝐾𝑋) = ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑋}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
50	44, 49	eleqtrrd 2865	1 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐾𝑋))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3086  Vcvv 3454  ∅c0 4285  {csn 4582  ⟨cop 4588  ∪ ciun 4949   class class class wbr 5100   × cxp 5645  ^◡ccnv 5646   “ cima 5650   Fn wfn 6516  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  Basecbs 17245  Hom chom 17297  Idccid 17697   Func cfunc 17887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  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 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810  df-ixp 8880  df-cat 17700  df-cid 17701  df-func 17891

This theorem is referenced by:  imasubc3  49777