Theorem tposresg 48751

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 6018	. . 3 ⊢ ((tpos 𝐹 ↾ ((V × V) ∪ {∅})) ↾ 𝑅) = ((tpos 𝐹 ↾ 𝑅) ↾ ((V × V) ∪ {∅}))
2		reltpos 8252	. . . . 5 ⊢ Rel tpos 𝐹
3		dmtposss 48749	. . . . 5 ⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})
4		relssres 6038	. . . . 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 5991	. . 3 ⊢ ((tpos 𝐹 ↾ ((V × V) ∪ {∅})) ↾ 𝑅) = (tpos 𝐹 ↾ 𝑅)
7		resres 6008	. . 3 ⊢ ((tpos 𝐹 ↾ 𝑅) ↾ ((V × V) ∪ {∅})) = (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅})))
8	1, 6, 7	3eqtr3i 2772	. 2 ⊢ (tpos 𝐹 ↾ 𝑅) = (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅})))
9		indi 4283	. . . 4 ⊢ (𝑅 ∩ ((V × V) ∪ {∅})) = ((𝑅 ∩ (V × V)) ∪ (𝑅 ∩ {∅}))
10		cnvcnv 6210	. . . . 5 ⊢ ◡◡𝑅 = (𝑅 ∩ (V × V))
11	10	uneq1i 4163	. . . 4 ⊢ (◡◡𝑅 ∪ (𝑅 ∩ {∅})) = ((𝑅 ∩ (V × V)) ∪ (𝑅 ∩ {∅}))
12	9, 11	eqtr4i 2767	. . 3 ⊢ (𝑅 ∩ ((V × V) ∪ {∅})) = (◡◡𝑅 ∪ (𝑅 ∩ {∅}))
13	12	reseq2i 5992	. 2 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅}))) = (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅})))
14		resundi 6009	. . 3 ⊢ (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (tpos 𝐹 ↾ (𝑅 ∩ {∅})))
15		rescom 6018	. . . . . 6 ⊢ ((tpos 𝐹 ↾ {∅}) ↾ 𝑅) = ((tpos 𝐹 ↾ 𝑅) ↾ {∅})
16		tposres0 48750	. . . . . . 7 ⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})
17	16	reseq1i 5991	. . . . . 6 ⊢ ((tpos 𝐹 ↾ {∅}) ↾ 𝑅) = ((𝐹 ↾ {∅}) ↾ 𝑅)
18		resres 6008	. . . . . 6 ⊢ ((tpos 𝐹 ↾ 𝑅) ↾ {∅}) = (tpos 𝐹 ↾ (𝑅 ∩ {∅}))
19	15, 17, 18	3eqtr3ri 2773	. . . . 5 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ {∅})) = ((𝐹 ↾ {∅}) ↾ 𝑅)
20		rescom 6018	. . . . 5 ⊢ ((𝐹 ↾ {∅}) ↾ 𝑅) = ((𝐹 ↾ 𝑅) ↾ {∅})
21		resres 6008	. . . . 5 ⊢ ((𝐹 ↾ 𝑅) ↾ {∅}) = (𝐹 ↾ (𝑅 ∩ {∅}))
22	19, 20, 21	3eqtri 2768	. . . 4 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ {∅})) = (𝐹 ↾ (𝑅 ∩ {∅}))
23	22	uneq2i 4164	. . 3 ⊢ ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (tpos 𝐹 ↾ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
24	14, 23	eqtri 2764	. 2 ⊢ (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
25	8, 13, 24	3eqtri 2768	1 ⊢ (tpos 𝐹 ↾ 𝑅) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))

Syntax hints:   = wceq 1540  Vcvv 3479   ∪ cun 3948   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  {csn 4624   × cxp 5681  ^◡ccnv 5682  dom cdm 5683   ↾ cres 5685  Rel wrel 5688  tpos ctpos 8246

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-fv 6567  df-tpos 8247

This theorem is referenced by:  tposres2  48753