Theorem tposres3 48842

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 48841	. 2 ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = (tpos 𝐹 ↾ ◡◡𝑅))
3		relcnv 6064	. . . . . . . 8 ⊢ Rel ◡dom (𝐹 ↾ ◡𝑅)
4		cnvf1o 8067	. . . . . . . 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 6789	. . . . . . 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 6757	. . . . . 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 6155	. . . . . 6 ⊢ ◡◡dom (𝐹 ↾ ◡𝑅) ⊆ dom (𝐹 ↾ ◡𝑅)
11		resdmss 6196	. . . . . 6 ⊢ dom (𝐹 ↾ ◡𝑅) ⊆ ◡𝑅
12	10, 11	sstri 3953	. . . . 5 ⊢ ◡◡dom (𝐹 ↾ ◡𝑅) ⊆ ◡𝑅
13	9, 12	eqsstri 3990	. . . 4 ⊢ ran (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}) ⊆ ◡𝑅
14		cores 6210	. . . 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 48836	. . . 4 ⊢ tpos (𝐹 ↾ ◡𝑅) = (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅})))
17		ressn 6246	. . . . . 6 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅}))
18		resres 5952	. . . . . . 7 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = (𝐹 ↾ (◡𝑅 ∩ {∅}))
19		relcnv 6064	. . . . . . . . . 10 ⊢ Rel ◡𝑅
20		0nelrel0 5691	. . . . . . . . . 10 ⊢ (Rel ◡𝑅 → ¬ ∅ ∈ ◡𝑅)
21	19, 20	ax-mp 5	. . . . . . . . 9 ⊢ ¬ ∅ ∈ ◡𝑅
22		disjsn 4671	. . . . . . . . 9 ⊢ ((◡𝑅 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡𝑅)
23	21, 22	mpbir 231	. . . . . . . 8 ⊢ (◡𝑅 ∩ {∅}) = ∅
24	23	reseq2i 5936	. . . . . . 7 ⊢ (𝐹 ↾ (◡𝑅 ∩ {∅})) = (𝐹 ↾ ∅)
25		res0 5943	. . . . . . 7 ⊢ (𝐹 ↾ ∅) = ∅
26	18, 24, 25	3eqtri 2756	. . . . . 6 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = ∅
27	17, 26	eqtr3i 2754	. . . . 5 ⊢ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅})) = ∅
28	27	uneq2i 4124	. . . 4 ⊢ (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅}))) = (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ∅)
29		un0 4353	. . . 4 ⊢ (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ∅) = ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
30	16, 28, 29	3eqtri 2756	. . 3 ⊢ tpos (𝐹 ↾ ◡𝑅) = ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
31		tposrescnv 48840	. . 3 ⊢ (tpos 𝐹 ↾ ◡◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
32	15, 30, 31	3eqtr4ri 2763	. 2 ⊢ (tpos 𝐹 ↾ ◡◡𝑅) = tpos (𝐹 ↾ ◡𝑅)
33	2, 32	eqtrdi 2780	1 ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = tpos (𝐹 ↾ ◡𝑅))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  {csn 4585  ∪ cuni 4867   ↦ cmpt 5183   × cxp 5629  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634   ∘ ccom 5635  Rel wrel 5636  –onto→wfo 6497  –1-1-onto→wf1o 6498  tpos ctpos 8181

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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-reu 3352  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-1st 7947  df-2nd 7948  df-tpos 8182

This theorem is referenced by:  tposres  48843