Theorem tposres3 50325

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 50324	. 2 ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = (tpos 𝐹 ↾ ◡◡𝑅))
3		relcnv 6118	. . . . . . . 8 ⊢ Rel ◡dom (𝐹 ↾ ◡𝑅)
4		cnvf1o 8159	. . . . . . . 8 ⊢ (Rel ◡dom (𝐹 ↾ ◡𝑅) → (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}):◡dom (𝐹 ↾ ◡𝑅)⟶1-1onto◡◡dom (𝐹 ↾ ◡𝑅))
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}):◡dom (𝐹 ↾ ◡𝑅)⟶1-1onto◡◡dom (𝐹 ↾ ◡𝑅)
6		f1ofo 6862	. . . . . . 7 ⊢ ((𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}):◡dom (𝐹 ↾ ◡𝑅)⟶1-1onto◡◡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 6829	. . . . . 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 6204	. . . . . 6 ⊢ ◡◡dom (𝐹 ↾ ◡𝑅) ⊆ dom (𝐹 ↾ ◡𝑅)
11		resdmss 6255	. . . . . 6 ⊢ dom (𝐹 ↾ ◡𝑅) ⊆ ◡𝑅
12	10, 11	sstri 3922	. . . . 5 ⊢ ◡◡dom (𝐹 ↾ ◡𝑅) ⊆ ◡𝑅
13	9, 12	eqsstri 3959	. . . 4 ⊢ ran (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}) ⊆ ◡𝑅
14		cores 6269	. . . 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 50319	. . . 4 ⊢ tpos (𝐹 ↾ ◡𝑅) = (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅})))
17		ressn 6312	. . . . . 6 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅}))
18		resres 5999	. . . . . . 7 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = (𝐹 ↾ (◡𝑅 ∩ {∅}))
19		relcnv 6118	. . . . . . . . . 10 ⊢ Rel ◡𝑅
20		0nelrel0 5718	. . . . . . . . . 10 ⊢ (Rel ◡𝑅 → ¬ ∅ ∈ ◡𝑅)
21	19, 20	ax-mp 5	. . . . . . . . 9 ⊢ ¬ ∅ ∈ ◡𝑅
22		disjsn 4673	. . . . . . . . 9 ⊢ ((◡𝑅 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ◡𝑅)
23	21, 22	mpbir 234	. . . . . . . 8 ⊢ (◡𝑅 ∩ {∅}) = ∅
24	23	reseq2i 5983	. . . . . . 7 ⊢ (𝐹 ↾ (◡𝑅 ∩ {∅})) = (𝐹 ↾ ∅)
25		res0 5990	. . . . . . 7 ⊢ (𝐹 ↾ ∅) = ∅
26	18, 24, 25	3eqtri 2790	. . . . . 6 ⊢ ((𝐹 ↾ ◡𝑅) ↾ {∅}) = ∅
27	17, 26	eqtr3i 2788	. . . . 5 ⊢ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅})) = ∅
28	27	uneq2i 4167	. . . 4 ⊢ (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ({∅} × ((𝐹 ↾ ◡𝑅) “ {∅}))) = (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ∅)
29		un0 4347	. . . 4 ⊢ (((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥})) ∪ ∅) = ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
30	16, 28, 29	3eqtri 2790	. . 3 ⊢ tpos (𝐹 ↾ ◡𝑅) = ((𝐹 ↾ ◡𝑅) ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
31		tposrescnv 50323	. . 3 ⊢ (tpos 𝐹 ↾ ◡◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ ◡𝑅) ↦ ∪ ◡{𝑥}))
32	15, 30, 31	3eqtr4ri 2797	. 2 ⊢ (tpos 𝐹 ↾ ◡◡𝑅) = tpos (𝐹 ↾ ◡𝑅)
33	2, 32	eqtrdi 2814	1 ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = tpos (𝐹 ↾ ◡𝑅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  ∪ cuni 4872   ↦ cmpt 5192   × cxp 5656  ^◡ccnv 5657  dom cdm 5658  ran crn 5659   ↾ cres 5660   “ cima 5661   ∘ ccom 5662  Rel wrel 5663  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565  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-reu 3370  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-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-1st 8035  df-2nd 8036  df-tpos 8281

This theorem is used by:  tposres  50326