Theorem imasubclem3 49943

Description: Lemma for imasubc 49988. (Contributed by Zhi Wang, 7-Nov-2025.)

Hypotheses

Ref	Expression
imasubclem1.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
imasubclem1.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)
imasubclem3.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
imasubclem3.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
imasubclem3.k	⊢ 𝐾 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ∪ 𝑧 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐺 “ {𝑦}))((𝐻‘𝐶) “ 𝐷))

Assertion

Ref	Expression
imasubclem3	⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝑦,𝐴,𝑥   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑦,𝐹,𝑧,𝑥   𝑦,𝐺,𝑧   𝑥,𝐻,𝑦   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)   𝐴(𝑧)   𝐵(𝑧)   𝐶(𝑧)   𝐷(𝑧)   𝐻(𝑧)   𝐾(𝑥, 𝑦, 𝑧)   𝑉(𝑥, 𝑦, 𝑧)   𝑊(𝑥, 𝑦, 𝑧)

Proof of Theorem imasubclem3

Step	Hyp	Ref	Expression
1		imasubclem3.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
2		imasubclem3.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
3		imasubclem1.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
4		imasubclem1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
5	3, 4	imasubclem1 49941	. 2 ⊢ (𝜑 → ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷) ∈ V)
6		simpl 488	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
7	6	sneqd 4603	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑥} = {𝑋})
8	7	imaeq2d 6064	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝑋}))
9		simpr 490	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
10	9	sneqd 4603	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑦} = {𝑌})
11	10	imaeq2d 6064	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {𝑌}))
12	8, 11	xpeq12d 5694	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((◡𝐹 “ {𝑥}) × (◡𝐺 “ {𝑦})) = ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌})))
13	12	iuneq1d 4986	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ∪ 𝑧 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐺 “ {𝑦}))((𝐻‘𝐶) “ 𝐷) = ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷))
14		imasubclem3.k	. . 3 ⊢ 𝐾 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ∪ 𝑧 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐺 “ {𝑦}))((𝐻‘𝐶) “ 𝐷))
15	13, 14	ovmpoga 7573	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷) ∈ V) → (𝑋𝐾𝑌) = ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷))
16	1, 2, 5, 15	syl3anc 1398	1 ⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591  ∪ ciun 4958   × cxp 5661  ^◡ccnv 5662   “ cima 5666  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424

This theorem is used by:  imaf1hom  49945  imasubc  49988  imassc  49990  imaid  49991