Theorem csbmpt2 5568

Description: Move substitution into the second part of a maps-to notation. (Contributed by AV, 26-Sep-2019.)

Assertion

Ref	Expression
csbmpt2	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑉   𝑦,𝑌,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)   𝑍(𝑥,𝑦)

Proof of Theorem csbmpt2

Step	Hyp	Ref	Expression
1		csbmpt12 5567	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍))
2		csbconstg 3927	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑌 = 𝑌)
3	2	mpteq1d 5243	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍))
4	1, 3	eqtrd 2775	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ⦋csb 3908   ↦ cmpt 5231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-mpt 5232

This theorem is referenced by:  matgsum  22459  csbrdgg  37312