Theorem csbmpt12 5438

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

Assertion

Ref	Expression
csbmpt12	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍))

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

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

Proof of Theorem csbmpt12

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbopab 5436	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
2		sbcan 3746	. . . . 5 ⊢ ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ∧ [𝐴 / 𝑥]𝑧 = 𝑍))
3		sbcel12 4323	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌)
4		csbconstg 3830	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
5	4	eleq1d 2822	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
6	3, 5	syl5bb 286	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
7		sbceq2g 4331	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑍 ↔ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍))
8	6, 7	anbi12d 634	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑦 ∈ 𝑌 ∧ [𝐴 / 𝑥]𝑧 = 𝑍) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)))
9	2, 8	syl5bb 286	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)))
10	9	opabbidv 5119	. . 3 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)})
11	1, 10	eqtrid 2789	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)})
12		df-mpt 5136	. . 3 ⊢ (𝑦 ∈ 𝑌 ↦ 𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
13	12	csbeq2i 3819	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
14		df-mpt 5136	. 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)}
15	11, 13, 14	3eqtr4g 2803	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  [wsbc 3694  ⦋csb 3811  {copab 5115   ↦ cmpt 5135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-opab 5116  df-mpt 5136

This theorem is referenced by:  csbmpt2  5439  esum2dlem  31772