Theorem csbmpt12 5506

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 5504	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
2		sbcan 3779	. . . . 5 ⊢ ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ∧ [𝐴 / 𝑥]𝑧 = 𝑍))
3		sbcel12 4346	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌)
4		csbconstg 3857	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
5	4	eleq1d 2825	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
6	3, 5	bitrid 284	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
7		sbceq2g 4354	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑍 ↔ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍))
8	6, 7	anbi12d 638	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑦 ∈ 𝑌 ∧ [𝐴 / 𝑥]𝑧 = 𝑍) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)))
9	2, 8	bitrid 284	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)))
10	9	opabbidv 5145	. . 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 2787	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)})
12		df-mpt 5161	. . 3 ⊢ (𝑦 ∈ 𝑌 ↦ 𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
13	12	csbeq2i 3846	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
14		df-mpt 5161	. 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)}
15	11, 13, 14	3eqtr4g 2800	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  [wsbc 3730  ⦋csb 3838  {copab 5141   ↦ cmpt 5160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-sn 4563  df-pr 4565  df-op 4569  df-opab 5142  df-mpt 5161

This theorem is referenced by:  csbmpt2  5507  esum2dlem  34283