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Theorem iunsnima 30385
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 3447 . . . 4 𝑥 ∈ V
2 vex 3447 . . . 4 𝑦 ∈ V
31, 2elimasn 5925 . . 3 (𝑦 ∈ ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
4 opeliunxp 5587 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
54baib 539 . . . 4 (𝑥𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ 𝑦𝐵))
65adantl 485 . . 3 ((𝜑𝑥𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ 𝑦𝐵))
73, 6syl5bb 286 . 2 ((𝜑𝑥𝐴) → (𝑦 ∈ ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ 𝑦𝐵))
87eqrdv 2799 1 ((𝜑𝑥𝐴) → ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  {csn 4528  cop 4534   ciun 4884   × cxp 5521  cima 5526
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-iun 4886  df-br 5034  df-opab 5096  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536
This theorem is referenced by:  esum2d  31460
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