Theorem tposres3 48857

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

Hypothesis

Ref	Expression
tposres2.1	⊢ (𝜑 → ¬ ∅ ∈ (dom 𝐹 ∩ 𝑅))

Assertion

Ref	Expression
tposres3	⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = tpos (𝐹 ↾ ◡𝑅))

Proof of Theorem tposres3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tposres2.1	. . 3 ⊢ (𝜑 → ¬ ∅ ∈ (dom 𝐹 ∩ 𝑅))
2	1	tposres2 48856	. 2 ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = (tpos 𝐹 ↾ ◡◡𝑅))
3		relcnv 6077	. . . . . . . 8 ⊢ Rel ◡dom (𝐹 ↾ ◡𝑅)
4		cnvf1o 8092	. . . . . . . 8 ⊢ (Rel ◡dom (𝐹 ↾ ◡𝑅) → (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}):◡dom (𝐹 ↾ ◡𝑅)–1-1-onto→◡◡dom (𝐹 ↾ ◡𝑅))
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}):◡dom (𝐹 ↾ ◡𝑅)–1-1-onto→◡◡dom (𝐹 ↾ ◡𝑅)
6		f1ofo 6809	. . . . . . 7 ⊢ ((𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}):◡dom (𝐹 ↾ ◡𝑅)–1-1-onto→◡◡dom (𝐹 ↾ ◡𝑅) → (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}):◡dom (𝐹 ↾ ◡𝑅)–onto→◡◡dom (𝐹 ↾ ◡𝑅))
7	5, 6	ax-mp 5	. . . . . 6 ⊢ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}):◡dom (𝐹 ↾ ◡𝑅)–onto→◡◡dom (𝐹 ↾ ◡𝑅)
8		forn 6777	. . . . . 6 ⊢ ((𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}):◡dom (𝐹 ↾ ◡𝑅)–onto→◡◡dom (𝐹 ↾ ◡𝑅) → ran (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}) = ◡◡dom (𝐹 ↾ ◡𝑅))
9	7, 8	ax-mp 5	. . . . 5 ⊢ ran (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}) = ◡◡dom (𝐹 ↾ ◡𝑅)
10		cnvcnvss 6169	. . . . . 6 ⊢ ◡◡dom (𝐹 ↾ ◡𝑅) ⊆ dom (𝐹 ↾ ◡𝑅)
11		resdmss 6210	. . . . . 6 ⊢ dom (𝐹 ↾ ◡𝑅) ⊆ ◡𝑅
12	10, 11	sstri 3958	. . . . 5 ⊢ ◡◡dom (𝐹 ↾ ◡𝑅) ⊆ ◡𝑅
13	9, 12	eqsstri 3995	. . . 4 ⊢ ran (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}) ⊆ ◡𝑅
14		cores 6224	. . . 4 ⊢ (ran (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}) ⊆ ◡𝑅 → ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})))
15	13, 14	ax-mp 5	. . 3 ⊢ ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
16		dftpos6 48851	. . . 4 ⊢ tpos (𝐹 ↾ ◡𝑅) = (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅})))
17		ressn 6260	. . . . . 6 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅}))
18		resres 5965	. . . . . . 7 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = (𝐹 ↾ (◡𝑅 ∩ {∅}))
19		relcnv 6077	. . . . . . . . . 10 ⊢ Rel ◡𝑅
20		0nelrel0 5700	. . . . . . . . . 10 ⊢ (Rel ◡𝑅 → ¬ ∅ ∈ ◡𝑅)
21	19, 20	ax-mp 5	. . . . . . . . 9 ⊢ ¬ ∅ ∈ ◡𝑅
22		disjsn 4677	. . . . . . . . 9 ⊢ ((◡𝑅 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡𝑅)
23	21, 22	mpbir 231	. . . . . . . 8 ⊢ (◡𝑅 ∩ {∅}) = ∅
24	23	reseq2i 5949	. . . . . . 7 ⊢ (𝐹 ↾ (◡𝑅 ∩ {∅})) = (𝐹 ↾ ∅)
25		res0 5956	. . . . . . 7 ⊢ (𝐹 ↾ ∅) = ∅
26	18, 24, 25	3eqtri 2757	. . . . . 6 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = ∅
27	17, 26	eqtr3i 2755	. . . . 5 ⊢ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅})) = ∅
28	27	uneq2i 4130	. . . 4 ⊢ (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅}))) = (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ∅)
29		un0 4359	. . . 4 ⊢ (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ∅) = ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
30	16, 28, 29	3eqtri 2757	. . 3 ⊢ tpos (𝐹 ↾ ◡𝑅) = ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
31		tposrescnv 48855	. . 3 ⊢ (tpos 𝐹 ↾ ◡◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
32	15, 30, 31	3eqtr4ri 2764	. 2 ⊢ (tpos 𝐹 ↾ ◡◡𝑅) = tpos (𝐹 ↾ ◡𝑅)
33	2, 32	eqtrdi 2781	1 ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = tpos (𝐹 ↾ ◡𝑅))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ∪ cun 3914   ∩ cin 3915   ⊆ wss 3916  ∅c0 4298  {csn 4591  ∪ cuni 4873   ↦ cmpt 5190   × cxp 5638  ^◡ccnv 5639  dom cdm 5640  ran crn 5641   ↾ cres 5642   “ cima 5643   ∘ ccom 5644  Rel wrel 5645  –onto→wfo 6511  –1-1-onto→wf1o 6512  tpos ctpos 8206

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 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-1st 7970  df-2nd 7971  df-tpos 8207

This theorem is referenced by:  tposres  48858