Theorem iunsnima 32546

Description: Image of a singleton by an indexed union involving that singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.)

Hypotheses

Ref	Expression
iunsnima.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
iunsnima.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)

Assertion

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

Proof of Theorem iunsnima

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3451	. . . 4 ⊢ 𝑥 ∈ V
2		vex 3451	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	elimasn 6061	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
4		opeliunxp 5705	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
5	4	baib 535	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 𝑦 ∈ 𝐵))
6	5	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 𝑦 ∈ 𝐵))
7	3, 6	bitrid 283	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ 𝑦 ∈ 𝐵))
8	7	eqrdv 2727	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4589  ⟨cop 4595  ∪ ciun 4955   × cxp 5636   “ cima 5641

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-iun 4957  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651

This theorem is referenced by:  esum2d  34083