Theorem imaf1hom 49914

Description: The hom-set of an image of a functor injective on objects. (Contributed by Zhi Wang, 7-Nov-2025.)

Hypotheses

Ref	Expression
imaf1hom.s	⊢ 𝑆 = (𝐹 “ 𝐴)
imaf1hom.1	⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)
imaf1hom.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)
imaf1hom.y	⊢ (𝜑 → 𝑌 ∈ 𝑆)
imaf1hom.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
imaf1hom.k	⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))

Assertion

Ref	Expression
imaf1hom	⊢ (𝜑 → (𝑋𝐾𝑌) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))

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

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑝)   𝐴(𝑥, 𝑦, 𝑝)   𝐵(𝑥, 𝑦, 𝑝)   𝐶(𝑥, 𝑦, 𝑝)   𝑆(𝑝)   𝐾(𝑥, 𝑦, 𝑝)   𝑉(𝑥, 𝑦, 𝑝)

Proof of Theorem imaf1hom

Step	Hyp	Ref	Expression
1		imaf1hom.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		imaf1hom.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
3		imaf1hom.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
4		imaf1hom.k	. . . 4 ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
5	1, 1, 2, 3, 4	imasubclem3 49912	. . 3 ⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌}))((𝐺‘𝑝) “ (𝐻‘𝑝)))
6		imaf1hom.s	. . . . . . . 8 ⊢ 𝑆 = (𝐹 “ 𝐴)
7		imaf1hom.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)
8	6, 7, 2	imaf1homlem 49913	. . . . . . 7 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))
9	8	simp1d 1160	. . . . . 6 ⊢ (𝜑 → {(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}))
10	6, 7, 3	imaf1homlem 49913	. . . . . . 7 ⊢ (𝜑 → ({(◡𝐹‘𝑌)} = (◡𝐹 “ {𝑌}) ∧ (𝐹‘(◡𝐹‘𝑌)) = 𝑌 ∧ (◡𝐹‘𝑌) ∈ 𝐵))
11	10	simp1d 1160	. . . . . 6 ⊢ (𝜑 → {(◡𝐹‘𝑌)} = (◡𝐹 “ {𝑌}))
12	9, 11	xpeq12d 5692	. . . . 5 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} × {(◡𝐹‘𝑌)}) = ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌})))
13		fvex 6894	. . . . . 6 ⊢ (◡𝐹‘𝑋) ∈ V
14		fvex 6894	. . . . . 6 ⊢ (◡𝐹‘𝑌) ∈ V
15	13, 14	xpsn 7137	. . . . 5 ⊢ ({(◡𝐹‘𝑋)} × {(◡𝐹‘𝑌)}) = {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩}
16	12, 15	eqtr3di 2813	. . . 4 ⊢ (𝜑 → ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌})) = {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩})
17	16	iuneq1d 4984	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌}))((𝐺‘𝑝) “ (𝐻‘𝑝)) = ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)))
18	5, 17	eqtrd 2798	. 2 ⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)))
19		opex 5445	. . 3 ⊢ ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ ∈ V
20		fveq2 6881	. . . . 5 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐺‘𝑝) = (𝐺‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩))
21		df-ov 7413	. . . . 5 ⊢ ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) = (𝐺‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩)
22	20, 21	eqtr4di 2816	. . . 4 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐺‘𝑝) = ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)))
23		fveq2 6881	. . . . 5 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐻‘𝑝) = (𝐻‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩))
24		df-ov 7413	. . . . 5 ⊢ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)) = (𝐻‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩)
25	23, 24	eqtr4di 2816	. . . 4 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐻‘𝑝) = ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))
26	22, 25	imaeq12d 6063	. . 3 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → ((𝐺‘𝑝) “ (𝐻‘𝑝)) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))
27	19, 26	iunxsn 5057	. 2 ⊢ ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))
28	18, 27	eqtrdi 2814	1 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2143  {csn 4589  ⟨cop 4595  ∪ ciun 4956   × cxp 5659  ^◡ccnv 5660   “ cima 5664  –1-1→wf1 6533  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  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-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415

This theorem is used by:  imaidfu  49916  imaf1co  49961