Theorem tposres3 49042

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 49041	. 2 ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = (tpos 𝐹 ↾ ◡◡𝑅))
3		relcnv 6060	. . . . . . . 8 ⊢ Rel ◡dom (𝐹 ↾ ◡𝑅)
4		cnvf1o 8050	. . . . . . . 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 6778	. . . . . . 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 6746	. . . . . 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 6149	. . . . . 6 ⊢ ◡◡dom (𝐹 ↾ ◡𝑅) ⊆ dom (𝐹 ↾ ◡𝑅)
11		resdmss 6190	. . . . . 6 ⊢ dom (𝐹 ↾ ◡𝑅) ⊆ ◡𝑅
12	10, 11	sstri 3940	. . . . 5 ⊢ ◡◡dom (𝐹 ↾ ◡𝑅) ⊆ ◡𝑅
13	9, 12	eqsstri 3977	. . . 4 ⊢ ran (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}) ⊆ ◡𝑅
14		cores 6204	. . . 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 49036	. . . 4 ⊢ tpos (𝐹 ↾ ◡𝑅) = (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅})))
17		ressn 6240	. . . . . 6 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅}))
18		resres 5948	. . . . . . 7 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = (𝐹 ↾ (◡𝑅 ∩ {∅}))
19		relcnv 6060	. . . . . . . . . 10 ⊢ Rel ◡𝑅
20		0nelrel0 5681	. . . . . . . . . 10 ⊢ (Rel ◡𝑅 → ¬ ∅ ∈ ◡𝑅)
21	19, 20	ax-mp 5	. . . . . . . . 9 ⊢ ¬ ∅ ∈ ◡𝑅
22		disjsn 4665	. . . . . . . . 9 ⊢ ((◡𝑅 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡𝑅)
23	21, 22	mpbir 231	. . . . . . . 8 ⊢ (◡𝑅 ∩ {∅}) = ∅
24	23	reseq2i 5932	. . . . . . 7 ⊢ (𝐹 ↾ (◡𝑅 ∩ {∅})) = (𝐹 ↾ ∅)
25		res0 5939	. . . . . . 7 ⊢ (𝐹 ↾ ∅) = ∅
26	18, 24, 25	3eqtri 2760	. . . . . 6 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = ∅
27	17, 26	eqtr3i 2758	. . . . 5 ⊢ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅})) = ∅
28	27	uneq2i 4114	. . . 4 ⊢ (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅}))) = (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ∅)
29		un0 4343	. . . 4 ⊢ (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ∅) = ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
30	16, 28, 29	3eqtri 2760	. . 3 ⊢ tpos (𝐹 ↾ ◡𝑅) = ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
31		tposrescnv 49040	. . 3 ⊢ (tpos 𝐹 ↾ ◡◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
32	15, 30, 31	3eqtr4ri 2767	. 2 ⊢ (tpos 𝐹 ↾ ◡◡𝑅) = tpos (𝐹 ↾ ◡𝑅)
33	2, 32	eqtrdi 2784	1 ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = tpos (𝐹 ↾ ◡𝑅))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2113   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  {csn 4577  ∪ cuni 4860   ↦ cmpt 5176   × cxp 5619  ^◡ccnv 5620  dom cdm 5621  ran crn 5622   ↾ cres 5623   “ cima 5624   ∘ ccom 5625  Rel wrel 5626  –onto→wfo 6487  –1-1-onto→wf1o 6488  tpos ctpos 8164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-1st 7930  df-2nd 7931  df-tpos 8165

This theorem is referenced by:  tposres  49043