Theorem iunsnima2 32554

Description: Version of iunsnima 32553 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 6063	. . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ ⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
2	1	elvd 3456	. . . 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 8192	. . . . 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 2728	1 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Ⅎwnfc 2877  Vcvv 3450  {csn 4592  ⟨cop 4598  ∪ ciun 4958   × cxp 5639   “ cima 5644

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-sep 5254  ax-nul 5264  ax-pr 5390

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-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-iun 4960  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654

This theorem is referenced by:  gsumpart  33004  gsumwrd2dccat  33014