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Theorem mptsnun 37059
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 4633 . . . . 5 (𝑥 = 𝑦 → {𝑥} = {𝑦})
21cbvmptv 5258 . . . 4 (𝑥𝐴 ↦ {𝑥}) = (𝑦𝐴 ↦ {𝑦})
32eqcomi 2735 . . 3 (𝑦𝐴 ↦ {𝑦}) = (𝑥𝐴 ↦ {𝑥})
4 mptsnun.r . . 3 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
53, 4mptsnunlem 37058 . 2 (𝐵𝐴𝐵 = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵))
6 mptsnun.f . . . . 5 𝐹 = (𝑥𝐴 ↦ {𝑥})
76, 2eqtri 2754 . . . 4 𝐹 = (𝑦𝐴 ↦ {𝑦})
87imaeq1i 6058 . . 3 (𝐹𝐵) = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵)
98unieqi 4917 . 2 (𝐹𝐵) = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵)
105, 9eqtr4di 2784 1 (𝐵𝐴𝐵 = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  {cab 2703  wrex 3060  wss 3946  {csn 4623   cuni 4905  cmpt 5228  cima 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-xp 5680  df-rel 5681  df-cnv 5682  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687
This theorem is referenced by:  dissneqlem  37060
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