Theorem imasubclem3 49596

Description: Lemma for imasubc 49641. (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 49594	. 2 ⊢ (𝜑 → ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷) ∈ V)
6		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
7	6	sneqd 4580	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑥} = {𝑋})
8	7	imaeq2d 6020	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝑋}))
9		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
10	9	sneqd 4580	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑦} = {𝑌})
11	10	imaeq2d 6020	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {𝑌}))
12	8, 11	xpeq12d 5656	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((◡𝐹 “ {𝑥}) × (◡𝐺 “ {𝑦})) = ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌})))
13	12	iuneq1d 4962	. . 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 7515	. 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 3430  {csn 4568  ∪ ciun 4934   × cxp 5623  ^◡ccnv 5624   “ cima 5628  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366

This theorem is referenced by:  imaf1hom  49598  imasubc  49641  imassc  49643  imaid  49644