Theorem tposresg 48872

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 5953	. . 3 ⊢ ((tpos 𝐹 ↾ ((V × V) ∪ {∅})) ↾ 𝑅) = ((tpos 𝐹 ↾ 𝑅) ↾ ((V × V) ∪ {∅}))
2		reltpos 8164	. . . . 5 ⊢ Rel tpos 𝐹
3		dmtposss 48870	. . . . 5 ⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})
4		relssres 5973	. . . . 5 ⊢ ((Rel tpos 𝐹 ∧ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})) → (tpos 𝐹 ↾ ((V × V) ∪ {∅})) = tpos 𝐹)
5	2, 3, 4	mp2an 692	. . . 4 ⊢ (tpos 𝐹 ↾ ((V × V) ∪ {∅})) = tpos 𝐹
6	5	reseq1i 5926	. . 3 ⊢ ((tpos 𝐹 ↾ ((V × V) ∪ {∅})) ↾ 𝑅) = (tpos 𝐹 ↾ 𝑅)
7		resres 5943	. . 3 ⊢ ((tpos 𝐹 ↾ 𝑅) ↾ ((V × V) ∪ {∅})) = (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅})))
8	1, 6, 7	3eqtr3i 2760	. 2 ⊢ (tpos 𝐹 ↾ 𝑅) = (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅})))
9		indi 4235	. . . 4 ⊢ (𝑅 ∩ ((V × V) ∪ {∅})) = ((𝑅 ∩ (V × V)) ∪ (𝑅 ∩ {∅}))
10		cnvcnv 6141	. . . . 5 ⊢ ◡◡𝑅 = (𝑅 ∩ (V × V))
11	10	uneq1i 4115	. . . 4 ⊢ (◡◡𝑅 ∪ (𝑅 ∩ {∅})) = ((𝑅 ∩ (V × V)) ∪ (𝑅 ∩ {∅}))
12	9, 11	eqtr4i 2755	. . 3 ⊢ (𝑅 ∩ ((V × V) ∪ {∅})) = (◡◡𝑅 ∪ (𝑅 ∩ {∅}))
13	12	reseq2i 5927	. 2 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅}))) = (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅})))
14		resundi 5944	. . 3 ⊢ (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (tpos 𝐹 ↾ (𝑅 ∩ {∅})))
15		rescom 5953	. . . . . 6 ⊢ ((tpos 𝐹 ↾ {∅}) ↾ 𝑅) = ((tpos 𝐹 ↾ 𝑅) ↾ {∅})
16		tposres0 48871	. . . . . . 7 ⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})
17	16	reseq1i 5926	. . . . . 6 ⊢ ((tpos 𝐹 ↾ {∅}) ↾ 𝑅) = ((𝐹 ↾ {∅}) ↾ 𝑅)
18		resres 5943	. . . . . 6 ⊢ ((tpos 𝐹 ↾ 𝑅) ↾ {∅}) = (tpos 𝐹 ↾ (𝑅 ∩ {∅}))
19	15, 17, 18	3eqtr3ri 2761	. . . . 5 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ {∅})) = ((𝐹 ↾ {∅}) ↾ 𝑅)
20		rescom 5953	. . . . 5 ⊢ ((𝐹 ↾ {∅}) ↾ 𝑅) = ((𝐹 ↾ 𝑅) ↾ {∅})
21		resres 5943	. . . . 5 ⊢ ((𝐹 ↾ 𝑅) ↾ {∅}) = (𝐹 ↾ (𝑅 ∩ {∅}))
22	19, 20, 21	3eqtri 2756	. . . 4 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ {∅})) = (𝐹 ↾ (𝑅 ∩ {∅}))
23	22	uneq2i 4116	. . 3 ⊢ ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (tpos 𝐹 ↾ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
24	14, 23	eqtri 2752	. 2 ⊢ (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
25	8, 13, 24	3eqtri 2756	1 ⊢ (tpos 𝐹 ↾ 𝑅) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))

Syntax hints:   = wceq 1540  Vcvv 3436   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  {csn 4577   × cxp 5617  ^◡ccnv 5618  dom cdm 5619   ↾ cres 5621  Rel wrel 5624  tpos ctpos 8158

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-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-fv 6490  df-tpos 8159

This theorem is referenced by:  tposres2  48874