Theorem imasubclem3 50361

Description: Lemma for imasubc 50406. (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 50359	. 2 ⊢ (𝜑 → ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷) ∈ V)
6		simpl 488	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
7	6	sneqd 4595	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑥} = {𝑋})
8	7	imaeq2d 6057	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝑋}))
9		simpr 490	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
10	9	sneqd 4595	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑦} = {𝑌})
11	10	imaeq2d 6057	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (◡𝐺 “ {𝑦}) = (◡𝐺 “ {𝑌}))
12	8, 11	xpeq12d 5685	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((◡𝐹 “ {𝑥}) × (◡𝐺 “ {𝑦})) = ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌})))
13	12	iuneq1d 4981	. . 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 7581	. 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 1569   ∈ wcel 2146  Vcvv 3453  {csn 4587  ∪ ciun 4953   × cxp 5652  ^◡ccnv 5653   “ cima 5657  ‘cfv 6546  (class class class)co 7427   ∈ cmpo 7429

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432

This theorem is used by:  imaf1hom  50363  imasubc  50406  imassc  50408  imaid  50409