Theorem imasubclem3 49581

Description: Lemma for imasubc 49626. (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 49579	. 2 ⊢ (𝜑 → ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷) ∈ V)
6		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
7	6	sneqd 4579	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑥} = {𝑋})
8	7	imaeq2d 6025	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝑋}))
9		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
10	9	sneqd 4579	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑦} = {𝑌})
11	10	imaeq2d 6025	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {𝑌}))
12	8, 11	xpeq12d 5662	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((◡𝐹 “ {𝑥}) × (◡𝐺 “ {𝑦})) = ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌})))
13	12	iuneq1d 4961	. . 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 7521	. 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 1374	1 ⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429  {csn 4567  ∪ ciun 4933   × cxp 5629  ^◡ccnv 5630   “ cima 5634  ‘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-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-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-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372

This theorem is referenced by:  imaf1hom  49583  imasubc  49626  imassc  49628  imaid  49629