Theorem csbmpt12 5446

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 5444	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
2		sbcan 3823	. . . . 5 ⊢ ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ∧ [𝐴 / 𝑥]𝑧 = 𝑍))
3		sbcel12 4362	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌)
4		csbconstg 3904	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
5	4	eleq1d 2899	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
6	3, 5	syl5bb 285	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
7		sbceq2g 4370	. . . . . 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	syl5bb 285	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)))
10	9	opabbidv 5134	. . 3 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)})
11	1, 10	syl5eq 2870	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)})
12		df-mpt 5149	. . 3 ⊢ (𝑦 ∈ 𝑌 ↦ 𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
13	12	csbeq2i 3893	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
14		df-mpt 5149	. 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)}
15	11, 13, 14	3eqtr4g 2883	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  [wsbc 3774  ⦋csb 3885  {copab 5130   ↦ cmpt 5148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-opab 5131  df-mpt 5149

This theorem is referenced by:  csbmpt2  5447  esum2dlem  31353