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Theorem iunsnima 30297
Description: Image of a singleton by an indexed union involving that singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.)
Hypotheses
Ref Expression
iunsnima.1 (𝜑𝐴𝑉)
iunsnima.2 ((𝜑𝑥𝐴) → 𝐵𝑊)
Assertion
Ref Expression
iunsnima ((𝜑𝑥𝐴) → ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵)

Proof of Theorem iunsnima
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3495 . . . 4 𝑥 ∈ V
2 vex 3495 . . . 4 𝑦 ∈ V
31, 2elimasn 5947 . . 3 (𝑦 ∈ ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
4 opeliunxp 5612 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
54baib 536 . . . 4 (𝑥𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ 𝑦𝐵))
65adantl 482 . . 3 ((𝜑𝑥𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ 𝑦𝐵))
73, 6syl5bb 284 . 2 ((𝜑𝑥𝐴) → (𝑦 ∈ ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ 𝑦𝐵))
87eqrdv 2816 1 ((𝜑𝑥𝐴) → ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {csn 4557  cop 4563   ciun 4910   × cxp 5546  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-iun 4912  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  esum2d  31251
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