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Theorem mptsnun 36024
Description: A class 𝐵 is equal to the union of the class of all singletons of elements of 𝐵. (Contributed by ML, 16-Jul-2020.)
Hypotheses
Ref Expression
mptsnun.f 𝐹 = (𝑥𝐴 ↦ {𝑥})
mptsnun.r 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
Assertion
Ref Expression
mptsnun (𝐵𝐴𝐵 = (𝐹𝐵))
Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵,𝑥
Allowed substitution hints:   𝑅(𝑥,𝑢)   𝐹(𝑥,𝑢)

Proof of Theorem mptsnun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sneq 4632 . . . . 5 (𝑥 = 𝑦 → {𝑥} = {𝑦})
21cbvmptv 5254 . . . 4 (𝑥𝐴 ↦ {𝑥}) = (𝑦𝐴 ↦ {𝑦})
32eqcomi 2740 . . 3 (𝑦𝐴 ↦ {𝑦}) = (𝑥𝐴 ↦ {𝑥})
4 mptsnun.r . . 3 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
53, 4mptsnunlem 36023 . 2 (𝐵𝐴𝐵 = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵))
6 mptsnun.f . . . . 5 𝐹 = (𝑥𝐴 ↦ {𝑥})
76, 2eqtri 2759 . . . 4 𝐹 = (𝑦𝐴 ↦ {𝑦})
87imaeq1i 6046 . . 3 (𝐹𝐵) = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵)
98unieqi 4914 . 2 (𝐹𝐵) = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵)
105, 9eqtr4di 2789 1 (𝐵𝐴𝐵 = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  {cab 2708  wrex 3069  wss 3944  {csn 4622   cuni 4901  cmpt 5224  cima 5672
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682
This theorem is referenced by:  dissneqlem  36025
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