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Theorem iunsnima 29412
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 3201 . . . 4 𝑥 ∈ V
2 vex 3201 . . . 4 𝑦 ∈ V
31, 2elimasn 5488 . . 3 (𝑦 ∈ ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
4 opeliunxp 5168 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
54baib 944 . . . 4 (𝑥𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ 𝑦𝐵))
65adantl 482 . . 3 ((𝜑𝑥𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ 𝑦𝐵))
73, 6syl5bb 272 . 2 ((𝜑𝑥𝐴) → (𝑦 ∈ ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ 𝑦𝐵))
87eqrdv 2619 1 ((𝜑𝑥𝐴) → ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  {csn 4175  cop 4181   ciun 4518   × cxp 5110  cima 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-iun 4520  df-br 4652  df-opab 4711  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125
This theorem is referenced by:  esum2d  30140
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