Theorem imaf1hom 49612

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 49610	. . 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 49611	. . . . . . 7 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))
9	8	simp1d 1149	. . . . . 6 ⊢ (𝜑 → {(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}))
10	6, 7, 3	imaf1homlem 49611	. . . . . . 7 ⊢ (𝜑 → ({(◡𝐹‘𝑌)} = (◡𝐹 “ {𝑌}) ∧ (𝐹‘(◡𝐹‘𝑌)) = 𝑌 ∧ (◡𝐹‘𝑌) ∈ 𝐵))
11	10	simp1d 1149	. . . . . 6 ⊢ (𝜑 → {(◡𝐹‘𝑌)} = (◡𝐹 “ {𝑌}))
12	9, 11	xpeq12d 5652	. . . . 5 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} × {(◡𝐹‘𝑌)}) = ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌})))
13		fvex 6844	. . . . . 6 ⊢ (◡𝐹‘𝑋) ∈ V
14		fvex 6844	. . . . . 6 ⊢ (◡𝐹‘𝑌) ∈ V
15	13, 14	xpsn 7087	. . . . 5 ⊢ ({(◡𝐹‘𝑋)} × {(◡𝐹‘𝑌)}) = {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩}
16	12, 15	eqtr3di 2791	. . . 4 ⊢ (𝜑 → ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌})) = {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩})
17	16	iuneq1d 4952	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌}))((𝐺‘𝑝) “ (𝐻‘𝑝)) = ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)))
18	5, 17	eqtrd 2776	. 2 ⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)))
19		opex 5406	. . 3 ⊢ ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ ∈ V
20		fveq2 6831	. . . . 5 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐺‘𝑝) = (𝐺‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩))
21		df-ov 7363	. . . . 5 ⊢ ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) = (𝐺‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩)
22	20, 21	eqtr4di 2794	. . . 4 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐺‘𝑝) = ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)))
23		fveq2 6831	. . . . 5 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐻‘𝑝) = (𝐻‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩))
24		df-ov 7363	. . . . 5 ⊢ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)) = (𝐻‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩)
25	23, 24	eqtr4di 2794	. . . 4 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐻‘𝑝) = ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))
26	22, 25	imaeq12d 6020	. . 3 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → ((𝐺‘𝑝) “ (𝐻‘𝑝)) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))
27	19, 26	iunxsn 5023	. 2 ⊢ ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))
28	18, 27	eqtrdi 2792	1 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  {csn 4558  ⟨cop 4564  ∪ ciun 4924   × cxp 5619  ^◡ccnv 5620   “ cima 5624  –1-1→wf1 6486  ‘cfv 6489  (class class class)co 7360   ∈ cmpo 7362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365

This theorem is referenced by:  imaidfu  49614  imaf1co  49659