Theorem iunsnima2 32708

Description: Version of iunsnima 32707 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 6056	. . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝑧 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) ↔ ⟨𝑌, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
2	1	elvd 3448	. . . 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 8162	. . . . 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 2735	1 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884  Vcvv 3442  {csn 4582  ⟨cop 4588  ∪ ciun 4948   × cxp 5630   “ cima 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-iun 4950  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645

This theorem is referenced by:  gsumpart  33156  gsumwrd2dccat  33171