Theorem iunsnima2 32602

Description: Version of iunsnima 32601 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 6037	. . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ ⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
2	1	elvd 3442	. . . 4 ⊢ (𝑌 ∈ 𝐴 → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ ⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
3	2	adantl 481	. . 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 8140	. . . . 5 ⊢ (⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑌 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))
7	6	baib 535	. . . 4 ⊢ (𝑌 ∈ 𝐴 → (⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 𝑧 ∈ 𝐶))
8	7	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 𝑧 ∈ 𝐶))
9	3, 8	bitrd 279	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ 𝑧 ∈ 𝐶))
10	9	eqrdv 2729	1 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Ⅎwnfc 2879  Vcvv 3436  {csn 4573  ⟨cop 4579  ∪ ciun 4939   × cxp 5612   “ cima 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-iun 4941  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627

This theorem is referenced by:  gsumpart  33037  gsumwrd2dccat  33047