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Theorem mptsnun 34704
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 4560 . . . . 5 (𝑥 = 𝑦 → {𝑥} = {𝑦})
21cbvmptv 5155 . . . 4 (𝑥𝐴 ↦ {𝑥}) = (𝑦𝐴 ↦ {𝑦})
32eqcomi 2833 . . 3 (𝑦𝐴 ↦ {𝑦}) = (𝑥𝐴 ↦ {𝑥})
4 mptsnun.r . . 3 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
53, 4mptsnunlem 34703 . 2 (𝐵𝐴𝐵 = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵))
6 mptsnun.f . . . . 5 𝐹 = (𝑥𝐴 ↦ {𝑥})
76, 2eqtri 2847 . . . 4 𝐹 = (𝑦𝐴 ↦ {𝑦})
87imaeq1i 5913 . . 3 (𝐹𝐵) = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵)
98unieqi 4837 . 2 (𝐹𝐵) = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵)
105, 9syl6eqr 2877 1 (𝐵𝐴𝐵 = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  {cab 2802  wrex 3134  wss 3919  {csn 4550   cuni 4824  cmpt 5132  cima 5545
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555
This theorem is referenced by:  dissneqlem  34705
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