tposideq - Mathbox for Zhi Wang

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Theorem tposideq 49546

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 49540	. 2 ⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = tpos ( I ↾ ^◡𝑅))
2		relcnv 6104	. . . . 5 ⊢ Rel ^◡𝑅
3		fnresi 6662	. . . . 5 ⊢ ( I ↾ ^◡𝑅) Fn ^◡𝑅
4		tposfn2 8240	. . . . 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 6185	. . . . . 6 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
7	6	biimpi 219	. . . . 5 ⊢ (Rel 𝑅 → ^◡^◡𝑅 = 𝑅)
8	7	fneq2d 6627	. . . 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 5404	. . . . . . 7 ⊢ {𝑥} ∈ V
11	10	cnvex 7918	. . . . . 6 ⊢ ^◡{𝑥} ∈ V
12	11	uniex 7736	. . . . 5 ⊢ ∪ ^◡{𝑥} ∈ V
13		eqid 2769	. . . . 5 ⊢ (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})
14	12, 13	fnmpti 6676	. . . 4 ⊢ (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) Fn 𝑅
15	14	a1i 11	. . 3 ⊢ (Rel 𝑅 → (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) Fn 𝑅)
16		1st2nd 8032	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
17		1st2ndb 8022	. . . . . 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 8031	. . . . 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 4601	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
22	21	cnveqd 5859	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ^◡{𝑥} = ^◡{𝑦})
23	22	unieqd 4886	. . . . . 6 ⊢ (𝑥 = 𝑦 → ∪ ^◡{𝑥} = ∪ ^◡{𝑦})
24	23, 13, 12	fvmpt3i 6993	. . . . 5 ⊢ (𝑦 ∈ 𝑅 → ((𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})‘𝑦) = ∪ ^◡{𝑦})
25	24	adantl 486	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → ((𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})‘𝑦) = ∪ ^◡{𝑦})
26	16	fveq2d 6883	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩))
27		ovtpos 8233	. . . . . . 7 ⊢ ((1^st ‘𝑦)tpos ( I ↾ ^◡𝑅)(2^nd ‘𝑦)) = ((2^nd ‘𝑦)( I ↾ ^◡𝑅)(1^st ‘𝑦))
28		df-ov 7411	. . . . . . 7 ⊢ ((1^st ‘𝑦)tpos ( I ↾ ^◡𝑅)(2^nd ‘𝑦)) = (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
29		df-ov 7411	. . . . . . 7 ⊢ ((2^nd ‘𝑦)( I ↾ ^◡𝑅)(1^st ‘𝑦)) = (( I ↾ ^◡𝑅)‘⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
30	27, 28, 29	3eqtr3i 2800	. . . . . 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 489	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝑦 ∈ 𝑅)
33	16, 32	eqeltrrd 2870	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ 𝑅)
34		fvex 6892	. . . . . . . 8 ⊢ (2^nd ‘𝑦) ∈ V
35		fvex 6892	. . . . . . . 8 ⊢ (1^st ‘𝑦) ∈ V
36	34, 35	opelcnv 5865	. . . . . . 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 7169	. . . . . 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 2808	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
41	20, 25, 40	3eqtr4rd 2815	. . 3 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = ((𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})‘𝑦))
42	9, 15, 41	eqfnfvd 7026	. 2 ⊢ (Rel 𝑅 → tpos ( I ↾ ^◡𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}))
43	1, 42	eqtrd 2804	1 ⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 Vcvv 3463 {csn 4591 ⟨cop 4597 ∪ cuni 4873 ↦ cmpt 5193 I cid 5553 × cxp 5657 ^◡ccnv 5658 ↾ cres 5661 Rel wrel 5664 Fn wfn 6529 ‘cfv 6534 (class class class)co 7408 1^st c1st 7980 2^nd c2nd 7981 tpos ctpos 8217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7411 df-1st 7982 df-2nd 7983 df-tpos 8218

This theorem is referenced by: tposideq2 49547

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