Theorem csbmpt12 5532

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 5530	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
2		sbcan 3815	. . . . 5 ⊢ ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ∧ [𝐴 / 𝑥]𝑧 = 𝑍))
3		sbcel12 4386	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌)
4		csbconstg 3893	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
5	4	eleq1d 2819	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
6	3, 5	bitrid 283	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
7		sbceq2g 4394	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑍 ↔ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍))
8	6, 7	anbi12d 632	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑦 ∈ 𝑌 ∧ [𝐴 / 𝑥]𝑧 = 𝑍) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)))
9	2, 8	bitrid 283	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)))
10	9	opabbidv 5185	. . 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 2782	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)})
12		df-mpt 5202	. . 3 ⊢ (𝑦 ∈ 𝑌 ↦ 𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
13	12	csbeq2i 3882	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
14		df-mpt 5202	. 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)}
15	11, 13, 14	3eqtr4g 2795	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  [wsbc 3765  ⦋csb 3874  {copab 5181   ↦ cmpt 5201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-opab 5182  df-mpt 5202

This theorem is referenced by:  csbmpt2  5533  esum2dlem  34123