Theorem mptsnun 33186

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 4187	. . . . 5 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
2	1	cbvmptv 4750	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑦 ∈ 𝐴 ↦ {𝑦})
3	2	eqcomi 2631	. . 3 ⊢ (𝑦 ∈ 𝐴 ↦ {𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
4		mptsnun.r	. . 3 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
5	3, 4	mptsnunlem 33185	. 2 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵))
6		mptsnun.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
7	6, 2	eqtri 2644	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ {𝑦})
8	7	imaeq1i 5463	. . 3 ⊢ (𝐹 “ 𝐵) = ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)
9	8	unieqi 4445	. 2 ⊢ ∪ (𝐹 “ 𝐵) = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)
10	5, 9	syl6eqr 2674	1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))

Syntax hints:   → wi 4   = wceq 1483  {cab 2608  ∃wrex 2913   ⊆ wss 3574  {csn 4177  ∪ cuni 4436   ↦ cmpt 4729   “ cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  dissneqlem  33187