Theorem parswapval 50867

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 2764	. . 3 ⊢ (𝑏 = 𝑈 → (𝑏 = 2o ↔ 𝑈 = 2o))
2		eqeq1 2764	. . . 4 ⊢ (𝑏 = 𝑈 → (𝑏 = ∅ ↔ 𝑈 = ∅))
3	2	ifbid 4505	. . 3 ⊢ (𝑏 = 𝑈 → if(𝑏 = ∅, 1o, ∅) = if(𝑈 = ∅, 1o, ∅))
4	1, 3	ifbieq2d 4508	. 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 8490	. . 3 ⊢ 2o ∈ V
7		1oex 8488	. . . 4 ⊢ 1o ∈ V
8		0ex 5260	. . . 4 ⊢ ∅ ∈ V
9	7, 8	ifex 4532	. . 3 ⊢ if(𝑈 = ∅, 1o, ∅) ∈ V
10	6, 9	ifex 4532	. 2 ⊢ if(𝑈 = 2o, 2o, if(𝑈 = ∅, 1o, ∅)) ∈ V
11	4, 5, 10	fvmpt 6993	1 ⊢ (𝑈 ∈ 3o → (𝐴‘𝑈) = if(𝑈 = 2o, 2o, if(𝑈 = ∅, 1o, ∅)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∅c0 4278  ifcif 4481   ↦ cmpt 5185  ‘cfv 6538  1_oc1o 8471  2_oc2o 8472  3_oc3o 8473

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-nul 5259  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-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-suc 6368  df-iota 6494  df-fun 6540  df-fv 6546  df-1o 8478  df-2o 8479

This theorem is used by:  parswapprop  50868