Theorem imasubclem3 49085

Description: Lemma for imasubc 49130. (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 49083	. 2 ⊢ (𝜑 → ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷) ∈ V)
6		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
7	6	sneqd 4603	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑥} = {𝑋})
8	7	imaeq2d 6033	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝑋}))
9		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
10	9	sneqd 4603	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑦} = {𝑌})
11	10	imaeq2d 6033	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {𝑌}))
12	8, 11	xpeq12d 5671	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((◡𝐹 “ {𝑥}) × (◡𝐺 “ {𝑦})) = ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌})))
13	12	iuneq1d 4985	. . 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 7545	. 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 1373	1 ⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  {csn 4591  ∪ ciun 4957   × cxp 5638  ^◡ccnv 5639   “ cima 5643  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394

This theorem is referenced by:  imaf1hom  49087  imasubc  49130  imassc  49132  imaid  49133