Theorem mptsnun 37372

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 = (𝑥 ∈ 𝐴 ↦ {𝑥})
mptsnun.r	⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}

Assertion

Ref	Expression
mptsnun	⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))

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.

Step	Hyp	Ref	Expression
1		sneq 4586	. . . . 5 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
2	1	cbvmptv 5195	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑦 ∈ 𝐴 ↦ {𝑦})
3	2	eqcomi 2740	. . 3 ⊢ (𝑦 ∈ 𝐴 ↦ {𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
4		mptsnun.r	. . 3 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
5	3, 4	mptsnunlem 37371	. 2 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵))
6		mptsnun.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
7	6, 2	eqtri 2754	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ {𝑦})
8	7	imaeq1i 6006	. . 3 ⊢ (𝐹 “ 𝐵) = ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)
9	8	unieqi 4871	. 2 ⊢ ∪ (𝐹 “ 𝐵) = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)
10	5, 9	eqtr4di 2784	1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))

Syntax hints:   → wi 4   = wceq 1541  {cab 2709  ∃wrex 3056   ⊆ wss 3902  {csn 4576  ∪ cuni 4859   ↦ cmpt 5172   “ cima 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629

This theorem is referenced by:  dissneqlem  37373