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Theorem mptsnun 33785
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 4408 . . . . 5 (𝑥 = 𝑦 → {𝑥} = {𝑦})
21cbvmptv 4987 . . . 4 (𝑥𝐴 ↦ {𝑥}) = (𝑦𝐴 ↦ {𝑦})
32eqcomi 2787 . . 3 (𝑦𝐴 ↦ {𝑦}) = (𝑥𝐴 ↦ {𝑥})
4 mptsnun.r . . 3 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
53, 4mptsnunlem 33784 . 2 (𝐵𝐴𝐵 = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵))
6 mptsnun.f . . . . 5 𝐹 = (𝑥𝐴 ↦ {𝑥})
76, 2eqtri 2802 . . . 4 𝐹 = (𝑦𝐴 ↦ {𝑦})
87imaeq1i 5719 . . 3 (𝐹𝐵) = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵)
98unieqi 4682 . 2 (𝐹𝐵) = ((𝑦𝐴 ↦ {𝑦}) “ 𝐵)
105, 9syl6eqr 2832 1 (𝐵𝐴𝐵 = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  {cab 2763  wrex 3091  wss 3792  {csn 4398   cuni 4673  cmpt 4967  cima 5360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-xp 5363  df-rel 5364  df-cnv 5365  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370
This theorem is referenced by:  dissneqlem  33786
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