Theorem tposrescnv 50323

Description: The transposition restricted to a converse is the transposition of the restricted class, with the empty set removed from the domain. Note that the right hand side is a more useful form of (tpos (𝐹 ↾ 𝑅) ↾ (V ∖ {∅})) by df-tpos 8281. (Contributed by Zhi Wang, 6-Oct-2025.)

Assertion

Ref	Expression
tposrescnv	⊢ (tpos 𝐹 ↾ ◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥}))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑅

Proof of Theorem tposrescnv

Step	Hyp	Ref	Expression
1		df-tpos 8281	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2	1	reseq1i 5982	. 2 ⊢ (tpos 𝐹 ↾ ◡𝑅) = ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡𝑅)
3		resco 6270	. 2 ⊢ ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡𝑅) = (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅))
4		resmpt3 6048	. . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅) = (𝑥 ∈ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) ↦ ∪ ◡{𝑥})
5		cnvin 6153	. . . . . 6 ⊢ ◡(𝑅 ∩ dom 𝐹) = (◡𝑅 ∩ ◡dom 𝐹)
6		dmres 6020	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝑅) = (𝑅 ∩ dom 𝐹)
7	6	cnveqi 5864	. . . . . 6 ⊢ ◡dom (𝐹 ↾ 𝑅) = ◡(𝑅 ∩ dom 𝐹)
8		incom 4109	. . . . . . 7 ⊢ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) = (◡𝑅 ∩ (◡dom 𝐹 ∪ {∅}))
9		indi 4232	. . . . . . 7 ⊢ (◡𝑅 ∩ (◡dom 𝐹 ∪ {∅})) = ((◡𝑅 ∩ ◡dom 𝐹) ∪ (◡𝑅 ∩ {∅}))
10		relcnv 6118	. . . . . . . . . . 11 ⊢ Rel ◡𝑅
11		0nelrel0 5718	. . . . . . . . . . 11 ⊢ (Rel ◡𝑅 → ¬ ∅ ∈ ◡𝑅)
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ ◡𝑅
13		disjsn 4673	. . . . . . . . . 10 ⊢ ((◡𝑅 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡𝑅)
14	12, 13	mpbir 234	. . . . . . . . 9 ⊢ (◡𝑅 ∩ {∅}) = ∅
15	14	uneq2i 4167	. . . . . . . 8 ⊢ ((◡𝑅 ∩ ◡dom 𝐹) ∪ (◡𝑅 ∩ {∅})) = ((◡𝑅 ∩ ◡dom 𝐹) ∪ ∅)
16		un0 4347	. . . . . . . 8 ⊢ ((◡𝑅 ∩ ◡dom 𝐹) ∪ ∅) = (◡𝑅 ∩ ◡dom 𝐹)
17	15, 16	eqtri 2786	. . . . . . 7 ⊢ ((◡𝑅 ∩ ◡dom 𝐹) ∪ (◡𝑅 ∩ {∅})) = (◡𝑅 ∩ ◡dom 𝐹)
18	8, 9, 17	3eqtri 2790	. . . . . 6 ⊢ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) = (◡𝑅 ∩ ◡dom 𝐹)
19	5, 7, 18	3eqtr4ri 2797	. . . . 5 ⊢ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) = ◡dom (𝐹 ↾ 𝑅)
20	19	mpteq1i 5202	. . . 4 ⊢ (𝑥 ∈ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥})
21	4, 20	eqtri 2786	. . 3 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅) = (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥})
22	21	coeq2i 5849	. 2 ⊢ (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅)) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥}))
23	2, 3, 22	3eqtri 2790	1 ⊢ (tpos 𝐹 ↾ ◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥}))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  ∪ cuni 4872   ↦ cmpt 5192  ^◡ccnv 5657  dom cdm 5658   ↾ cres 5660   ∘ ccom 5662  Rel wrel 5663  tpos ctpos 8280

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-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

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-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  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 4841  df-br 5109  df-opab 5174  df-mpt 5193  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-res 5670  df-tpos 8281

This theorem is used by:  tposres3  50325