Theorem imaf1hom 50216

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 50214	. . 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 50215	. . . . . . 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 50215	. . . . . . 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 5681	. . . . 5 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} × {(◡𝐹‘𝑌)}) = ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌})))
13		fvex 6898	. . . . . 6 ⊢ (◡𝐹‘𝑋) ∈ V
14		fvex 6898	. . . . . 6 ⊢ (◡𝐹‘𝑌) ∈ V
15	13, 14	xpsn 7141	. . . . 5 ⊢ ({(◡𝐹‘𝑋)} × {(◡𝐹‘𝑌)}) = {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩}
16	12, 15	eqtr3di 2810	. . . 4 ⊢ (𝜑 → ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌})) = {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩})
17	16	iuneq1d 4978	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌}))((𝐺‘𝑝) “ (𝐻‘𝑝)) = ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)))
18	5, 17	eqtrd 2795	. 2 ⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)))
19		opex 5431	. . 3 ⊢ ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ ∈ V
20		fveq2 6885	. . . . 5 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐺‘𝑝) = (𝐺‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩))
21		df-ov 7423	. . . . 5 ⊢ ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) = (𝐺‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩)
22	20, 21	eqtr4di 2813	. . . 4 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐺‘𝑝) = ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)))
23		fveq2 6885	. . . . 5 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐻‘𝑝) = (𝐻‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩))
24		df-ov 7423	. . . . 5 ⊢ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)) = (𝐻‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩)
25	23, 24	eqtr4di 2813	. . . 4 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐻‘𝑝) = ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))
26	22, 25	imaeq12d 6052	. . 3 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → ((𝐺‘𝑝) “ (𝐻‘𝑝)) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))
27	19, 26	iunxsn 5050	. 2 ⊢ ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))
28	18, 27	eqtrdi 2811	1 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {csn 4583  ⟨cop 4589  ∪ ciun 4950   × cxp 5648  ^◡ccnv 5649   “ cima 5653  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

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 2099  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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  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 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is used by:  imaidfu  50218  imaf1co  50263