Theorem ifmpt2v 7494

Description: Move a conditional inside and outside a function in maps-to notation. (Contributed by SN, 16-Oct-2025.)

Assertion

Ref	Expression
ifmpt2v	⊢ (𝑥 ∈ 𝐴 ↦ if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝑥 ∈ 𝐴 ↦ 𝐵), (𝑥 ∈ 𝐴 ↦ 𝐶))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ifmpt2v

Step	Hyp	Ref	Expression
1		iftrue 4497	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐶) = 𝐵)
2	1	mpteq2dv 5204	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝜑, 𝐵, 𝐶)) = (𝑥 ∈ 𝐴 ↦ 𝐵))
3		iftrue 4497	. . 3 ⊢ (𝜑 → if(𝜑, (𝑥 ∈ 𝐴 ↦ 𝐵), (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ 𝐵))
4	2, 3	eqtr4d 2768	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝑥 ∈ 𝐴 ↦ 𝐵), (𝑥 ∈ 𝐴 ↦ 𝐶)))
5		iffalse 4500	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐶) = 𝐶)
6	5	mpteq2dv 5204	. . 3 ⊢ (¬ 𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝜑, 𝐵, 𝐶)) = (𝑥 ∈ 𝐴 ↦ 𝐶))
7		iffalse 4500	. . 3 ⊢ (¬ 𝜑 → if(𝜑, (𝑥 ∈ 𝐴 ↦ 𝐵), (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ 𝐶))
8	6, 7	eqtr4d 2768	. 2 ⊢ (¬ 𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝑥 ∈ 𝐴 ↦ 𝐵), (𝑥 ∈ 𝐴 ↦ 𝐶)))
9	4, 8	pm2.61i 182	1 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝑥 ∈ 𝐴 ↦ 𝐵), (𝑥 ∈ 𝐴 ↦ 𝐶))

Syntax hints:  ¬ wn 3   = wceq 1540  ifcif 4491   ↦ cmpt 5191

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-if 4492  df-opab 5173  df-mpt 5192

This theorem is referenced by:  psdmvr  22063