Theorem tposrescnv 48840

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 8182. (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 8182	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2	1	reseq1i 5935	. 2 ⊢ (tpos 𝐹 ↾ ◡𝑅) = ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡𝑅)
3		resco 6211	. 2 ⊢ ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡𝑅) = (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅))
4		resmpt3 5998	. . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅) = (𝑥 ∈ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) ↦ ∪ ◡{𝑥})
5		cnvin 6105	. . . . . 6 ⊢ ◡(𝑅 ∩ dom 𝐹) = (◡𝑅 ∩ ◡dom 𝐹)
6		dmres 5972	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝑅) = (𝑅 ∩ dom 𝐹)
7	6	cnveqi 5828	. . . . . 6 ⊢ ◡dom (𝐹 ↾ 𝑅) = ◡(𝑅 ∩ dom 𝐹)
8		incom 4168	. . . . . . 7 ⊢ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) = (◡𝑅 ∩ (◡dom 𝐹 ∪ {∅}))
9		indi 4243	. . . . . . 7 ⊢ (◡𝑅 ∩ (◡dom 𝐹 ∪ {∅})) = ((◡𝑅 ∩ ◡dom 𝐹) ∪ (◡𝑅 ∩ {∅}))
10		relcnv 6064	. . . . . . . . . . 11 ⊢ Rel ◡𝑅
11		0nelrel0 5691	. . . . . . . . . . 11 ⊢ (Rel ◡𝑅 → ¬ ∅ ∈ ◡𝑅)
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ ◡𝑅
13		disjsn 4671	. . . . . . . . . 10 ⊢ ((◡𝑅 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡𝑅)
14	12, 13	mpbir 231	. . . . . . . . 9 ⊢ (◡𝑅 ∩ {∅}) = ∅
15	14	uneq2i 4124	. . . . . . . 8 ⊢ ((◡𝑅 ∩ ◡dom 𝐹) ∪ (◡𝑅 ∩ {∅})) = ((◡𝑅 ∩ ◡dom 𝐹) ∪ ∅)
16		un0 4353	. . . . . . . 8 ⊢ ((◡𝑅 ∩ ◡dom 𝐹) ∪ ∅) = (◡𝑅 ∩ ◡dom 𝐹)
17	15, 16	eqtri 2752	. . . . . . 7 ⊢ ((◡𝑅 ∩ ◡dom 𝐹) ∪ (◡𝑅 ∩ {∅})) = (◡𝑅 ∩ ◡dom 𝐹)
18	8, 9, 17	3eqtri 2756	. . . . . 6 ⊢ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) = (◡𝑅 ∩ ◡dom 𝐹)
19	5, 7, 18	3eqtr4ri 2763	. . . . 5 ⊢ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) = ◡dom (𝐹 ↾ 𝑅)
20	19	mpteq1i 5193	. . . 4 ⊢ (𝑥 ∈ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥})
21	4, 20	eqtri 2752	. . 3 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅) = (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥})
22	21	coeq2i 5814	. 2 ⊢ (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅)) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥}))
23	2, 3, 22	3eqtri 2756	1 ⊢ (tpos 𝐹 ↾ ◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥}))

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109   ∪ cun 3909   ∩ cin 3910  ∅c0 4292  {csn 4585  ∪ cuni 4867   ↦ cmpt 5183  ^◡ccnv 5630  dom cdm 5631   ↾ cres 5633   ∘ ccom 5635  Rel wrel 5636  tpos ctpos 8181

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-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-tpos 8182

This theorem is referenced by:  tposres3  48842