Theorem iunsnima2 33001

Description: Version of iunsnima 33000 with different variables. (Contributed by Thierry Arnoux, 22-Jun-2024.)

Hypotheses

Ref	Expression
iunsnima.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
iunsnima.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
iunsnima2.1	⊢ Ⅎ𝑥𝐶
iunsnima2.2	⊢ (𝑥 = 𝑌 → 𝐵 = 𝐶)

Assertion

Ref	Expression
iunsnima2	⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑌

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem iunsnima2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elimasng 6096	. . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ ⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
2	1	elvd 3464	. . . 4 ⊢ (𝑌 ∈ 𝐴 → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ ⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
3	2	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ ⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
4		iunsnima2.1	. . . . . 6 ⊢ Ⅎ𝑥𝐶
5		iunsnima2.2	. . . . . 6 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐶)
6	4, 5	opeliunxp2f 8215	. . . . 5 ⊢ (⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑌 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))
7	6	baib 545	. . . 4 ⊢ (𝑌 ∈ 𝐴 → (⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 𝑧 ∈ 𝐶))
8	7	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 𝑧 ∈ 𝐶))
9	3, 8	bitrd 282	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ 𝑧 ∈ 𝐶))
10	9	eqrdv 2764	1 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Ⅎwnfc 2913  Vcvv 3458  {csn 4594  ⟨cop 4600  ∪ ciun 4961   × cxp 5664   “ cima 5669

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 2738  ax-sep 5262  ax-pr 5409

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-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-iun 4963  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679

This theorem is used by:  gsumpart  33414  gsumwrd2dccat  33429