Theorem mptsnun 37797

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 4591	. . . . 5 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
2	1	cbvmptv 5203	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑦 ∈ 𝐴 ↦ {𝑦})
3	2	eqcomi 2770	. . 3 ⊢ (𝑦 ∈ 𝐴 ↦ {𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑥})
4		mptsnun.r	. . 3 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
5	3, 4	mptsnunlem 37796	. 2 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵))
6		mptsnun.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})
7	6, 2	eqtri 2784	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ {𝑦})
8	7	imaeq1i 6043	. . 3 ⊢ (𝐹 “ 𝐵) = ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)
9	8	unieqi 4876	. 2 ⊢ ∪ (𝐹 “ 𝐵) = ∪ ((𝑦 ∈ 𝐴 ↦ {𝑦}) “ 𝐵)
10	5, 9	eqtr4di 2814	1 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))

Syntax hints:   → wi 4   = wceq 1559  {cab 2739  ∃wrex 3085   ⊆ wss 3904  {csn 4581  ∪ cuni 4864   ↦ cmpt 5180   “ cima 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-xp 5651  df-rel 5652  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658

This theorem is referenced by:  dissneqlem  37798