Theorem imaf1hom 49583

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 49581	. . 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 49582	. . . . . . 7 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))
9	8	simp1d 1143	. . . . . 6 ⊢ (𝜑 → {(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}))
10	6, 7, 3	imaf1homlem 49582	. . . . . . 7 ⊢ (𝜑 → ({(◡𝐹‘𝑌)} = (◡𝐹 “ {𝑌}) ∧ (𝐹‘(◡𝐹‘𝑌)) = 𝑌 ∧ (◡𝐹‘𝑌) ∈ 𝐵))
11	10	simp1d 1143	. . . . . 6 ⊢ (𝜑 → {(◡𝐹‘𝑌)} = (◡𝐹 “ {𝑌}))
12	9, 11	xpeq12d 5662	. . . . 5 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} × {(◡𝐹‘𝑌)}) = ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌})))
13		fvex 6853	. . . . . 6 ⊢ (◡𝐹‘𝑋) ∈ V
14		fvex 6853	. . . . . 6 ⊢ (◡𝐹‘𝑌) ∈ V
15	13, 14	xpsn 7094	. . . . 5 ⊢ ({(◡𝐹‘𝑋)} × {(◡𝐹‘𝑌)}) = {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩}
16	12, 15	eqtr3di 2786	. . . 4 ⊢ (𝜑 → ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌})) = {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩})
17	16	iuneq1d 4961	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌}))((𝐺‘𝑝) “ (𝐻‘𝑝)) = ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)))
18	5, 17	eqtrd 2771	. 2 ⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)))
19		opex 5416	. . 3 ⊢ ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ ∈ V
20		fveq2 6840	. . . . 5 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐺‘𝑝) = (𝐺‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩))
21		df-ov 7370	. . . . 5 ⊢ ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) = (𝐺‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩)
22	20, 21	eqtr4di 2789	. . . 4 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐺‘𝑝) = ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)))
23		fveq2 6840	. . . . 5 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐻‘𝑝) = (𝐻‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩))
24		df-ov 7370	. . . . 5 ⊢ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)) = (𝐻‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩)
25	23, 24	eqtr4di 2789	. . . 4 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐻‘𝑝) = ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))
26	22, 25	imaeq12d 6026	. . 3 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → ((𝐺‘𝑝) “ (𝐻‘𝑝)) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))
27	19, 26	iunxsn 5033	. 2 ⊢ ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))
28	18, 27	eqtrdi 2787	1 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4567  ⟨cop 4573  ∪ ciun 4933   × cxp 5629  ^◡ccnv 5630   “ cima 5634  –1-1→wf1 6495  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372

This theorem is referenced by:  imaidfu  49585  imaf1co  49630