Theorem tposres3 49356

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 49355	. 2 ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = (tpos 𝐹 ↾ ◡◡𝑅))
3		relcnv 6069	. . . . . . . 8 ⊢ Rel ◡dom (𝐹 ↾ ◡𝑅)
4		cnvf1o 8061	. . . . . . . 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 6787	. . . . . . 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 6755	. . . . . 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 6158	. . . . . 6 ⊢ ◡◡dom (𝐹 ↾ ◡𝑅) ⊆ dom (𝐹 ↾ ◡𝑅)
11		resdmss 6199	. . . . . 6 ⊢ dom (𝐹 ↾ ◡𝑅) ⊆ ◡𝑅
12	10, 11	sstri 3931	. . . . 5 ⊢ ◡◡dom (𝐹 ↾ ◡𝑅) ⊆ ◡𝑅
13	9, 12	eqsstri 3968	. . . 4 ⊢ ran (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}) ⊆ ◡𝑅
14		cores 6213	. . . 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 49350	. . . 4 ⊢ tpos (𝐹 ↾ ◡𝑅) = (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅})))
17		ressn 6249	. . . . . 6 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅}))
18		resres 5957	. . . . . . 7 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = (𝐹 ↾ (◡𝑅 ∩ {∅}))
19		relcnv 6069	. . . . . . . . . 10 ⊢ Rel ◡𝑅
20		0nelrel0 5691	. . . . . . . . . 10 ⊢ (Rel ◡𝑅 → ¬ ∅ ∈ ◡𝑅)
21	19, 20	ax-mp 5	. . . . . . . . 9 ⊢ ¬ ∅ ∈ ◡𝑅
22		disjsn 4655	. . . . . . . . 9 ⊢ ((◡𝑅 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡𝑅)
23	21, 22	mpbir 231	. . . . . . . 8 ⊢ (◡𝑅 ∩ {∅}) = ∅
24	23	reseq2i 5941	. . . . . . 7 ⊢ (𝐹 ↾ (◡𝑅 ∩ {∅})) = (𝐹 ↾ ∅)
25		res0 5948	. . . . . . 7 ⊢ (𝐹 ↾ ∅) = ∅
26	18, 24, 25	3eqtri 2763	. . . . . 6 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = ∅
27	17, 26	eqtr3i 2761	. . . . 5 ⊢ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅})) = ∅
28	27	uneq2i 4105	. . . 4 ⊢ (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅}))) = (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ∅)
29		un0 4334	. . . 4 ⊢ (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ∅) = ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
30	16, 28, 29	3eqtri 2763	. . 3 ⊢ tpos (𝐹 ↾ ◡𝑅) = ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
31		tposrescnv 49354	. . 3 ⊢ (tpos 𝐹 ↾ ◡◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
32	15, 30, 31	3eqtr4ri 2770	. 2 ⊢ (tpos 𝐹 ↾ ◡◡𝑅) = tpos (𝐹 ↾ ◡𝑅)
33	2, 32	eqtrdi 2787	1 ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = tpos (𝐹 ↾ ◡𝑅))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  {csn 4567  ∪ cuni 4850   ↦ cmpt 5166   × cxp 5629  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634   ∘ ccom 5635  Rel wrel 5636  –onto→wfo 6496  –1-1-onto→wf1o 6497  tpos ctpos 8175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-1st 7942  df-2nd 7943  df-tpos 8176

This theorem is referenced by:  tposres  49357