Theorem iunsnima 32544

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 3463	. . . 4 ⊢ 𝑥 ∈ V
2		vex 3463	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	elimasn 6077	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
4		opeliunxp 5721	. . . . 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 2733	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {csn 4601  ⟨cop 4607  ∪ ciun 4967   × cxp 5652   “ cima 5657

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-iun 4969  df-br 5120  df-opab 5182  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667

This theorem is referenced by:  esum2d  34070