Theorem tposresg 48747

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 5987	. . 3 ⊢ ((tpos 𝐹 ↾ ((V × V) ∪ {∅})) ↾ 𝑅) = ((tpos 𝐹 ↾ 𝑅) ↾ ((V × V) ∪ {∅}))
2		reltpos 8225	. . . . 5 ⊢ Rel tpos 𝐹
3		dmtposss 48745	. . . . 5 ⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})
4		relssres 6007	. . . . 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 5960	. . 3 ⊢ ((tpos 𝐹 ↾ ((V × V) ∪ {∅})) ↾ 𝑅) = (tpos 𝐹 ↾ 𝑅)
7		resres 5977	. . 3 ⊢ ((tpos 𝐹 ↾ 𝑅) ↾ ((V × V) ∪ {∅})) = (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅})))
8	1, 6, 7	3eqtr3i 2765	. 2 ⊢ (tpos 𝐹 ↾ 𝑅) = (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅})))
9		indi 4257	. . . 4 ⊢ (𝑅 ∩ ((V × V) ∪ {∅})) = ((𝑅 ∩ (V × V)) ∪ (𝑅 ∩ {∅}))
10		cnvcnv 6179	. . . . 5 ⊢ ◡◡𝑅 = (𝑅 ∩ (V × V))
11	10	uneq1i 4137	. . . 4 ⊢ (◡◡𝑅 ∪ (𝑅 ∩ {∅})) = ((𝑅 ∩ (V × V)) ∪ (𝑅 ∩ {∅}))
12	9, 11	eqtr4i 2760	. . 3 ⊢ (𝑅 ∩ ((V × V) ∪ {∅})) = (◡◡𝑅 ∪ (𝑅 ∩ {∅}))
13	12	reseq2i 5961	. 2 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅}))) = (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅})))
14		resundi 5978	. . 3 ⊢ (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (tpos 𝐹 ↾ (𝑅 ∩ {∅})))
15		rescom 5987	. . . . . 6 ⊢ ((tpos 𝐹 ↾ {∅}) ↾ 𝑅) = ((tpos 𝐹 ↾ 𝑅) ↾ {∅})
16		tposres0 48746	. . . . . . 7 ⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})
17	16	reseq1i 5960	. . . . . 6 ⊢ ((tpos 𝐹 ↾ {∅}) ↾ 𝑅) = ((𝐹 ↾ {∅}) ↾ 𝑅)
18		resres 5977	. . . . . 6 ⊢ ((tpos 𝐹 ↾ 𝑅) ↾ {∅}) = (tpos 𝐹 ↾ (𝑅 ∩ {∅}))
19	15, 17, 18	3eqtr3ri 2766	. . . . 5 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ {∅})) = ((𝐹 ↾ {∅}) ↾ 𝑅)
20		rescom 5987	. . . . 5 ⊢ ((𝐹 ↾ {∅}) ↾ 𝑅) = ((𝐹 ↾ 𝑅) ↾ {∅})
21		resres 5977	. . . . 5 ⊢ ((𝐹 ↾ 𝑅) ↾ {∅}) = (𝐹 ↾ (𝑅 ∩ {∅}))
22	19, 20, 21	3eqtri 2761	. . . 4 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ {∅})) = (𝐹 ↾ (𝑅 ∩ {∅}))
23	22	uneq2i 4138	. . 3 ⊢ ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (tpos 𝐹 ↾ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
24	14, 23	eqtri 2757	. 2 ⊢ (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
25	8, 13, 24	3eqtri 2761	1 ⊢ (tpos 𝐹 ↾ 𝑅) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))

Syntax hints:   = wceq 1539  Vcvv 3457   ∪ cun 3922   ∩ cin 3923   ⊆ wss 3924  ∅c0 4306  {csn 4599   × cxp 5650  ^◡ccnv 5651  dom cdm 5652   ↾ cres 5654  Rel wrel 5657  tpos ctpos 8219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-fv 6536  df-tpos 8220

This theorem is referenced by:  tposres2  48749