Theorem tposrescnv 49003

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 8162. (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 8162	. . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
2	1	reseq1i 5928	. 2 ⊢ (tpos 𝐹 ↾ ◡𝑅) = ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡𝑅)
3		resco 6202	. 2 ⊢ ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡𝑅) = (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅))
4		resmpt3 5991	. . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅) = (𝑥 ∈ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) ↦ ∪ ◡{𝑥})
5		cnvin 6096	. . . . . 6 ⊢ ◡(𝑅 ∩ dom 𝐹) = (◡𝑅 ∩ ◡dom 𝐹)
6		dmres 5965	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝑅) = (𝑅 ∩ dom 𝐹)
7	6	cnveqi 5818	. . . . . 6 ⊢ ◡dom (𝐹 ↾ 𝑅) = ◡(𝑅 ∩ dom 𝐹)
8		incom 4158	. . . . . . 7 ⊢ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) = (◡𝑅 ∩ (◡dom 𝐹 ∪ {∅}))
9		indi 4233	. . . . . . 7 ⊢ (◡𝑅 ∩ (◡dom 𝐹 ∪ {∅})) = ((◡𝑅 ∩ ◡dom 𝐹) ∪ (◡𝑅 ∩ {∅}))
10		relcnv 6057	. . . . . . . . . . 11 ⊢ Rel ◡𝑅
11		0nelrel0 5679	. . . . . . . . . . 11 ⊢ (Rel ◡𝑅 → ¬ ∅ ∈ ◡𝑅)
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ ◡𝑅
13		disjsn 4663	. . . . . . . . . 10 ⊢ ((◡𝑅 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡𝑅)
14	12, 13	mpbir 231	. . . . . . . . 9 ⊢ (◡𝑅 ∩ {∅}) = ∅
15	14	uneq2i 4114	. . . . . . . 8 ⊢ ((◡𝑅 ∩ ◡dom 𝐹) ∪ (◡𝑅 ∩ {∅})) = ((◡𝑅 ∩ ◡dom 𝐹) ∪ ∅)
16		un0 4343	. . . . . . . 8 ⊢ ((◡𝑅 ∩ ◡dom 𝐹) ∪ ∅) = (◡𝑅 ∩ ◡dom 𝐹)
17	15, 16	eqtri 2756	. . . . . . 7 ⊢ ((◡𝑅 ∩ ◡dom 𝐹) ∪ (◡𝑅 ∩ {∅})) = (◡𝑅 ∩ ◡dom 𝐹)
18	8, 9, 17	3eqtri 2760	. . . . . 6 ⊢ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) = (◡𝑅 ∩ ◡dom 𝐹)
19	5, 7, 18	3eqtr4ri 2767	. . . . 5 ⊢ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) = ◡dom (𝐹 ↾ 𝑅)
20	19	mpteq1i 5184	. . . 4 ⊢ (𝑥 ∈ ((◡dom 𝐹 ∪ {∅}) ∩ ◡𝑅) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥})
21	4, 20	eqtri 2756	. . 3 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅) = (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥})
22	21	coeq2i 5804	. 2 ⊢ (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡𝑅)) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥}))
23	2, 3, 22	3eqtri 2760	1 ⊢ (tpos 𝐹 ↾ ◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥}))

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113   ∪ cun 3896   ∩ cin 3897  ∅c0 4282  {csn 4575  ∪ cuni 4858   ↦ cmpt 5174  ^◡ccnv 5618  dom cdm 5619   ↾ cres 5621   ∘ ccom 5623  Rel wrel 5624  tpos ctpos 8161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-mpt 5175  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-res 5631  df-tpos 8162

This theorem is referenced by:  tposres3  49005