Theorem mptsnun 37307

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 4658	. . . . 5 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
2	1	cbvmptv 5279	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑦 ∈ 𝐴 ↦ {𝑦})
3	2	eqcomi 2749	. . 3 ⊢ (𝑦 ∈ 𝐴 ↦ {𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
4		mptsnun.r	. . 3 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
5	3, 4	mptsnunlem 37306	. 2 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵))
6		mptsnun.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
7	6, 2	eqtri 2768	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ {𝑦})
8	7	imaeq1i 6088	. . 3 ⊢ (𝐹 “ 𝐵) = ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)
9	8	unieqi 4943	. 2 ⊢ ∪ (𝐹 “ 𝐵) = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)
10	5, 9	eqtr4di 2798	1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))

Syntax hints:   → wi 4   = wceq 1537  {cab 2717  ∃wrex 3076   ⊆ wss 3976  {csn 4648  ∪ cuni 4931   ↦ cmpt 5249   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  dissneqlem  37308