Theorem csbmpt12 5531

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 5529	. . 3 ⊢ ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
2		sbcan 3787	. . . . 5 ⊢ ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ∧ [𝐴 / 𝑥]𝑧 = 𝑍))
3		sbcel12 4368	. . . . . . 7 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌)
4		csbconstg 3865	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
5	4	eleq1d 2845	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
6	3, 5	bitrid 286	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝑌 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌))
7		sbceq2g 4376	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑍 ↔ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍))
8	6, 7	anbi12d 644	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑦 ∈ 𝑌 ∧ [𝐴 / 𝑥]𝑧 = 𝑍) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)))
9	2, 8	bitrid 286	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)))
10	9	opabbidv 5170	. . 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 2807	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)})
12		df-mpt 5186	. . 3 ⊢ (𝑦 ∈ 𝑌 ↦ 𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
13	12	csbeq2i 3854	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝑍)}
14		df-mpt 5186	. 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ∧ 𝑧 = ⦋𝐴 / 𝑥⦌𝑍)}
15	11, 13, 14	3eqtr4g 2820	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  [wsbc 3738  ⦋csb 3846  {copab 5166   ↦ cmpt 5185

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-sn 4585  df-pr 4587  df-op 4835  df-opab 5167  df-mpt 5186

This theorem is used by:  csbmpt2  5532  esum2dlem  34771