Theorem parswapval 50981

Description: Evaluate the transposition of the two parallel-arrow values. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypothesis

Ref	Expression
parswapval.a	⊢ 𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))

Assertion

Ref	Expression
parswapval	⊢ (𝑈 ∈ 3o → (𝐴‘𝑈) = if(𝑈 = 2o, 2o, if(𝑈 = ∅, 1o, ∅)))

Distinct variable group:   𝑈,𝑏

Allowed substitution hint:   𝐴(𝑏)

Proof of Theorem parswapval

Step	Hyp	Ref	Expression
1		eqeq1 2767	. . 3 ⊢ (𝑏 = 𝑈 → (𝑏 = 2o ↔ 𝑈 = 2o))
2		eqeq1 2767	. . . 4 ⊢ (𝑏 = 𝑈 → (𝑏 = ∅ ↔ 𝑈 = ∅))
3	2	ifbid 4509	. . 3 ⊢ (𝑏 = 𝑈 → if(𝑏 = ∅, 1o, ∅) = if(𝑈 = ∅, 1o, ∅))
4	1, 3	ifbieq2d 4512	. 2 ⊢ (𝑏 = 𝑈 → if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)) = if(𝑈 = 2o, 2o, if(𝑈 = ∅, 1o, ∅)))
5		parswapval.a	. 2 ⊢ 𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))
6		2oex 8516	. . 3 ⊢ 2o ∈ V
7		1oex 8514	. . . 4 ⊢ 1o ∈ V
8		0ex 5264	. . . 4 ⊢ ∅ ∈ V
9	7, 8	ifex 4536	. . 3 ⊢ if(𝑈 = ∅, 1o, ∅) ∈ V
10	6, 9	ifex 4536	. 2 ⊢ if(𝑈 = 2o, 2o, if(𝑈 = ∅, 1o, ∅)) ∈ V
11	4, 5, 10	fvmpt 7016	1 ⊢ (𝑈 ∈ 3o → (𝐴‘𝑈) = if(𝑈 = 2o, 2o, if(𝑈 = ∅, 1o, ∅)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∅c0 4281  ifcif 4485   ↦ cmpt 5189  ‘cfv 6558  1_oc1o 8497  2_oc2o 8498  3_oc3o 8499

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-suc 6386  df-iota 6512  df-fun 6560  df-fv 6566  df-1o 8504  df-2o 8505

This theorem is used by:  parswapprop  50982