Theorem tposresg 50322

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 6009	. . 3 ⊢ ((tpos 𝐹 ↾ ((V × V) ∪ {∅})) ↾ 𝑅) = ((tpos 𝐹 ↾ 𝑅) ↾ ((V × V) ∪ {∅}))
2		reltpos 8286	. . . . 5 ⊢ Rel tpos 𝐹
3		dmtposss 50320	. . . . 5 ⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})
4		relssres 6029	. . . . 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 5982	. . 3 ⊢ ((tpos 𝐹 ↾ ((V × V) ∪ {∅})) ↾ 𝑅) = (tpos 𝐹 ↾ 𝑅)
7		resres 5999	. . 3 ⊢ ((tpos 𝐹 ↾ 𝑅) ↾ ((V × V) ∪ {∅})) = (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅})))
8	1, 6, 7	3eqtr3i 2794	. 2 ⊢ (tpos 𝐹 ↾ 𝑅) = (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅})))
9		indi 4232	. . . 4 ⊢ (𝑅 ∩ ((V × V) ∪ {∅})) = ((𝑅 ∩ (V × V)) ∪ (𝑅 ∩ {∅}))
10		cnvcnv 6202	. . . . 5 ⊢ ◡◡𝑅 = (𝑅 ∩ (V × V))
11	10	uneq1i 4166	. . . 4 ⊢ (◡◡𝑅 ∪ (𝑅 ∩ {∅})) = ((𝑅 ∩ (V × V)) ∪ (𝑅 ∩ {∅}))
12	9, 11	eqtr4i 2789	. . 3 ⊢ (𝑅 ∩ ((V × V) ∪ {∅})) = (◡◡𝑅 ∪ (𝑅 ∩ {∅}))
13	12	reseq2i 5983	. 2 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ ((V × V) ∪ {∅}))) = (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅})))
14		resundi 6000	. . 3 ⊢ (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (tpos 𝐹 ↾ (𝑅 ∩ {∅})))
15		rescom 6009	. . . . . 6 ⊢ ((tpos 𝐹 ↾ {∅}) ↾ 𝑅) = ((tpos 𝐹 ↾ 𝑅) ↾ {∅})
16		tposres0 50321	. . . . . . 7 ⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})
17	16	reseq1i 5982	. . . . . 6 ⊢ ((tpos 𝐹 ↾ {∅}) ↾ 𝑅) = ((𝐹 ↾ {∅}) ↾ 𝑅)
18		resres 5999	. . . . . 6 ⊢ ((tpos 𝐹 ↾ 𝑅) ↾ {∅}) = (tpos 𝐹 ↾ (𝑅 ∩ {∅}))
19	15, 17, 18	3eqtr3ri 2795	. . . . 5 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ {∅})) = ((𝐹 ↾ {∅}) ↾ 𝑅)
20		rescom 6009	. . . . 5 ⊢ ((𝐹 ↾ {∅}) ↾ 𝑅) = ((𝐹 ↾ 𝑅) ↾ {∅})
21		resres 5999	. . . . 5 ⊢ ((𝐹 ↾ 𝑅) ↾ {∅}) = (𝐹 ↾ (𝑅 ∩ {∅}))
22	19, 20, 21	3eqtri 2790	. . . 4 ⊢ (tpos 𝐹 ↾ (𝑅 ∩ {∅})) = (𝐹 ↾ (𝑅 ∩ {∅}))
23	22	uneq2i 4167	. . 3 ⊢ ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (tpos 𝐹 ↾ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
24	14, 23	eqtri 2786	. 2 ⊢ (tpos 𝐹 ↾ (◡◡𝑅 ∪ (𝑅 ∩ {∅}))) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
25	8, 13, 24	3eqtri 2790	1 ⊢ (tpos 𝐹 ↾ 𝑅) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588   × cxp 5656  ^◡ccnv 5657  dom cdm 5658   ↾ cres 5660  Rel wrel 5663  tpos ctpos 8280

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-fv 6574  df-tpos 8281

This theorem is used by:  tposres2  50324