Theorem tposresg 49715

Description: The transposition restricted to a set. (Contributed by Zhi Wang, 6-Oct-2025.)

Assertion

Ref	Expression
tposresg	⊢ (tpos 𝐹 ↾ 𝑅) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))

Proof of Theorem tposresg

Step	Hyp	Ref	Expression
1		rescom 6003	. . 3 ⊢ ((tpos 𝐹 ↾ ((V × V) ∪ {∅})) ↾ 𝑅) = ((tpos 𝐹 ↾ 𝑅) ↾ ((V × V) ∪ {∅}))
2		reltpos 8233	. . . . 5 ⊢ Rel tpos 𝐹
3		dmtposss 49713	. . . . 5 ⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})
4		relssres 6023	. . . . 5 ⊢ ((Rel tpos 𝐹 ∧ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})) → (tpos 𝐹 ↾ ((V × V) ∪ {∅})) = tpos 𝐹)
5	2, 3, 4	mp2an 705	. . . 4 ⊢ (tpos 𝐹 ↾ ((V × V) ∪ {∅})) = tpos 𝐹
6	5	reseq1i 5976	. . 3 ⊢ ((tpos 𝐹 ↾ ((V × V) ∪ {∅})) ↾ 𝑅) = (tpos 𝐹 ↾ 𝑅)
7		resres 5993	. . 3 ⊢ ((tpos 𝐹 ↾ 𝑅) ↾ ((V × V) ∪ {∅})) = (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅})))
8	1, 6, 7	3eqtr3i 2796	. 2 ⊢ (tpos 𝐹 ↾ 𝑅) = (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅})))
9		indi 4237	. . . 4 ⊢ (𝑅 ∩ ((V × V) ∪ {∅})) = ((𝑅 ∩ (V × V)) ∪ (𝑅 ∩ {∅}))
10		cnvcnv 6192	. . . . 5 ⊢ ◡◡𝑅 = (𝑅 ∩ (V × V))
11	10	uneq1i 4118	. . . 4 ⊢ (◡◡𝑅 ∪ (𝑅 ∩ {∅})) = ((𝑅 ∩ (V × V)) ∪ (𝑅 ∩ {∅}))
12	9, 11	eqtr4i 2791	. . 3 ⊢ (𝑅 ∩ ((V × V) ∪ {∅})) = (◡◡𝑅 ∪ (𝑅 ∩ {∅}))
13	12	reseq2i 5977	. 2 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅}))) = (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅})))
14		resundi 5994	. . 3 ⊢ (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (tpos 𝐹 ↾ (𝑅 ∩ {∅})))
15		rescom 6003	. . . . . 6 ⊢ ((tpos 𝐹 ↾ {∅}) ↾ 𝑅) = ((tpos 𝐹 ↾ 𝑅) ↾ {∅})
16		tposres0 49714	. . . . . . 7 ⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})
17	16	reseq1i 5976	. . . . . 6 ⊢ ((tpos 𝐹 ↾ {∅}) ↾ 𝑅) = ((𝐹 ↾ {∅}) ↾ 𝑅)
18		resres 5993	. . . . . 6 ⊢ ((tpos 𝐹 ↾ 𝑅) ↾ {∅}) = (tpos 𝐹 ↾ (𝑅 ∩ {∅}))
19	15, 17, 18	3eqtr3ri 2797	. . . . 5 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ {∅})) = ((𝐹 ↾ {∅}) ↾ 𝑅)
20		rescom 6003	. . . . 5 ⊢ ((𝐹 ↾ {∅}) ↾ 𝑅) = ((𝐹 ↾ 𝑅) ↾ {∅})
21		resres 5993	. . . . 5 ⊢ ((𝐹 ↾ 𝑅) ↾ {∅}) = (𝐹 ↾ (𝑅 ∩ {∅}))
22	19, 20, 21	3eqtri 2792	. . . 4 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ {∅})) = (𝐹 ↾ (𝑅 ∩ {∅}))
23	22	uneq2i 4119	. . 3 ⊢ ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (tpos 𝐹 ↾ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
24	14, 23	eqtri 2788	. 2 ⊢ (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
25	8, 13, 24	3eqtri 2792	1 ⊢ (tpos 𝐹 ↾ 𝑅) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3457   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591   × cxp 5661  ^◡ccnv 5662  dom cdm 5663   ↾ cres 5665  Rel wrel 5668  tpos ctpos 8227

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-tpos 8228

This theorem is used by:  tposres2  49717