tposideq - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > tposideq		Structured version Visualization version GIF version

Theorem tposideq 50332

Description: Two ways of expressing the swap function. (Contributed by Zhi Wang, 6-Oct-2025.)

**Assertion**
Ref	Expression
tposideq	⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}))

Distinct variable group: 𝑥,𝑅

Proof of Theorem tposideq Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tposres 50326	. 2 ⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = tpos ( I ↾ ^◡𝑅))
2		relcnv 6118	. . . . 5 ⊢ Rel ^◡𝑅
3		fnresi 6698	. . . . 5 ⊢ ( I ↾ ^◡𝑅) Fn ^◡𝑅
4		tposfn2 8303	. . . . 5 ⊢ (Rel ^◡𝑅 → (( I ↾ ^◡𝑅) Fn ^◡𝑅 → tpos ( I ↾ ^◡𝑅) Fn ^◡^◡𝑅))
5	2, 3, 4	mp2 9	. . . 4 ⊢ tpos ( I ↾ ^◡𝑅) Fn ^◡^◡𝑅
6		dfrel2 6199	. . . . . 6 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
7	6	biimpi 219	. . . . 5 ⊢ (Rel 𝑅 → ^◡^◡𝑅 = 𝑅)
8	7	fneq2d 6663	. . . 4 ⊢ (Rel 𝑅 → (tpos ( I ↾ ^◡𝑅) Fn ^◡^◡𝑅 ↔ tpos ( I ↾ ^◡𝑅) Fn 𝑅))
9	5, 8	mpbii 236	. . 3 ⊢ (Rel 𝑅 → tpos ( I ↾ ^◡𝑅) Fn 𝑅)
10		vsnex 5400	. . . . . . 7 ⊢ {𝑥} ∈ V
11	10	cnvex 7971	. . . . . 6 ⊢ ^◡{𝑥} ∈ V
12	11	uniex 7790	. . . . 5 ⊢ ∪ ^◡{𝑥} ∈ V
13		eqid 2763	. . . . 5 ⊢ (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})
14	12, 13	fnmpti 6712	. . . 4 ⊢ (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) Fn 𝑅
15	14	a1i 11	. . 3 ⊢ (Rel 𝑅 → (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) Fn 𝑅)
16		1st2nd 8084	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
17		1st2ndb 8075	. . . . . 6 ⊢ (𝑦 ∈ (V × V) ↔ 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
18	17	biimpri 231	. . . . 5 ⊢ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → 𝑦 ∈ (V × V))
19		2nd1st 8083	. . . . 5 ⊢ (𝑦 ∈ (V × V) → ∪ ^◡{𝑦} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
20	16, 18, 19	3syl 19	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → ∪ ^◡{𝑦} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
21		sneq 4594	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
22	21	cnveqd 5865	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ^◡{𝑥} = ^◡{𝑦})
23	22	unieqd 4885	. . . . . 6 ⊢ (𝑥 = 𝑦 → ∪ ^◡{𝑥} = ∪ ^◡{𝑦})
24	23, 13, 12	fvmpt3i 7030	. . . . 5 ⊢ (𝑦 ∈ 𝑅 → ((𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})‘𝑦) = ∪ ^◡{𝑦})
25	24	adantl 487	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → ((𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})‘𝑦) = ∪ ^◡{𝑦})
26	16	fveq2d 6920	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩))
27		ovtpos 8296	. . . . . . 7 ⊢ ((1^st ‘𝑦)tpos ( I ↾ ^◡𝑅)(2^nd ‘𝑦)) = ((2^nd ‘𝑦)( I ↾ ^◡𝑅)(1^st ‘𝑦))
28		df-ov 7454	. . . . . . 7 ⊢ ((1^st ‘𝑦)tpos ( I ↾ ^◡𝑅)(2^nd ‘𝑦)) = (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
29		df-ov 7454	. . . . . . 7 ⊢ ((2^nd ‘𝑦)( I ↾ ^◡𝑅)(1^st ‘𝑦)) = (( I ↾ ^◡𝑅)‘⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
30	27, 28, 29	3eqtr3i 2794	. . . . . 6 ⊢ (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩) = (( I ↾ ^◡𝑅)‘⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
31	30	a1i 11	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩) = (( I ↾ ^◡𝑅)‘⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩))
32		simpr 490	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝑦 ∈ 𝑅)
33	16, 32	eqeltrrd 2865	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ 𝑅)
34		fvex 6929	. . . . . . . 8 ⊢ (2^nd ‘𝑦) ∈ V
35		fvex 6929	. . . . . . . 8 ⊢ (1^st ‘𝑦) ∈ V
36	34, 35	opelcnv 5871	. . . . . . 7 ⊢ (⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ ∈ ^◡𝑅 ↔ ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ 𝑅)
37	36	biimpri 231	. . . . . 6 ⊢ (⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ 𝑅 → ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ ∈ ^◡𝑅)
38		fvresi 7209	. . . . . 6 ⊢ (⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ ∈ ^◡𝑅 → (( I ↾ ^◡𝑅)‘⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
39	33, 37, 38	3syl 19	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (( I ↾ ^◡𝑅)‘⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
40	26, 31, 39	3eqtrd 2802	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
41	20, 25, 40	3eqtr4rd 2809	. . 3 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = ((𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})‘𝑦))
42	9, 15, 41	eqfnfvd 7063	. 2 ⊢ (Rel 𝑅 → tpos ( I ↾ ^◡𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}))
43	1, 42	eqtrd 2798	1 ⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 Vcvv 3454 {csn 4588 ⟨cop 4840 ∪ cuni 4872 ↦ cmpt 5192 I cid 5552 × cxp 5656 ^◡ccnv 5657 ↾ cres 5660 Rel wrel 5663 Fn wfn 6561 ‘cfv 6566 (class class class)co 7451 1^st c1st 8033 2^nd c2nd 8034 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-sbc 3743 df-csb 3852 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-ov 7454 df-1st 8035 df-2nd 8036 df-tpos 8281

This theorem is used by: tposideq2 50333

W3C validator