Theorem imaf1hom 49808

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 49806	. . 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 49807	. . . . . . 7 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))
9	8	simp1d 1158	. . . . . 6 ⊢ (𝜑 → {(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}))
10	6, 7, 3	imaf1homlem 49807	. . . . . . 7 ⊢ (𝜑 → ({(◡𝐹‘𝑌)} = (◡𝐹 “ {𝑌}) ∧ (𝐹‘(◡𝐹‘𝑌)) = 𝑌 ∧ (◡𝐹‘𝑌) ∈ 𝐵))
11	10	simp1d 1158	. . . . . 6 ⊢ (𝜑 → {(◡𝐹‘𝑌)} = (◡𝐹 “ {𝑌}))
12	9, 11	xpeq12d 5695	. . . . 5 ⊢ (𝜑 → ({(◡𝐹‘𝑋)} × {(◡𝐹‘𝑌)}) = ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌})))
13		fvex 6897	. . . . . 6 ⊢ (◡𝐹‘𝑋) ∈ V
14		fvex 6897	. . . . . 6 ⊢ (◡𝐹‘𝑌) ∈ V
15	13, 14	xpsn 7140	. . . . 5 ⊢ ({(◡𝐹‘𝑋)} × {(◡𝐹‘𝑌)}) = {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩}
16	12, 15	eqtr3di 2819	. . . 4 ⊢ (𝜑 → ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌})) = {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩})
17	16	iuneq1d 4988	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐹 “ {𝑌}))((𝐺‘𝑝) “ (𝐻‘𝑝)) = ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)))
18	5, 17	eqtrd 2804	. 2 ⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)))
19		opex 5448	. . 3 ⊢ ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ ∈ V
20		fveq2 6884	. . . . 5 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐺‘𝑝) = (𝐺‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩))
21		df-ov 7416	. . . . 5 ⊢ ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) = (𝐺‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩)
22	20, 21	eqtr4di 2822	. . . 4 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐺‘𝑝) = ((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)))
23		fveq2 6884	. . . . 5 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐻‘𝑝) = (𝐻‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩))
24		df-ov 7416	. . . . 5 ⊢ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)) = (𝐻‘⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩)
25	23, 24	eqtr4di 2822	. . . 4 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → (𝐻‘𝑝) = ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))
26	22, 25	imaeq12d 6066	. . 3 ⊢ (𝑝 = ⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩ → ((𝐺‘𝑝) “ (𝐻‘𝑝)) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))
27	19, 26	iunxsn 5061	. 2 ⊢ ∪ 𝑝 ∈ {⟨(◡𝐹‘𝑋), (◡𝐹‘𝑌)⟩} ((𝐺‘𝑝) “ (𝐻‘𝑝)) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))
28	18, 27	eqtrdi 2820	1 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {csn 4594  ⟨cop 4600  ∪ ciun 4960   × cxp 5662  ^◡ccnv 5663   “ cima 5667  –1-1→wf1 6536  ‘cfv 6539  (class class class)co 7413   ∈ cmpo 7415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7416  df-oprab 7417  df-mpo 7418

This theorem is referenced by:  imaidfu  49810  imaf1co  49855