tposideq - Mathbox for Zhi Wang

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Theorem tposideq 48864

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 48858	. 2 ⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = tpos ( I ↾ ^◡𝑅))
2		relcnv 6077	. . . . 5 ⊢ Rel ^◡𝑅
3		fnresi 6649	. . . . 5 ⊢ ( I ↾ ^◡𝑅) Fn ^◡𝑅
4		tposfn2 8229	. . . . 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 6164	. . . . . 6 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
7	6	biimpi 216	. . . . 5 ⊢ (Rel 𝑅 → ^◡^◡𝑅 = 𝑅)
8	7	fneq2d 6614	. . . 4 ⊢ (Rel 𝑅 → (tpos ( I ↾ ^◡𝑅) Fn ^◡^◡𝑅 ↔ tpos ( I ↾ ^◡𝑅) Fn 𝑅))
9	5, 8	mpbii 233	. . 3 ⊢ (Rel 𝑅 → tpos ( I ↾ ^◡𝑅) Fn 𝑅)
10		vsnex 5391	. . . . . . 7 ⊢ {𝑥} ∈ V
11	10	cnvex 7903	. . . . . 6 ⊢ ^◡{𝑥} ∈ V
12	11	uniex 7719	. . . . 5 ⊢ ∪ ^◡{𝑥} ∈ V
13		eqid 2730	. . . . 5 ⊢ (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})
14	12, 13	fnmpti 6663	. . . 4 ⊢ (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) Fn 𝑅
15	14	a1i 11	. . 3 ⊢ (Rel 𝑅 → (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) Fn 𝑅)
16		1st2nd 8020	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
17		1st2ndb 8010	. . . . . 6 ⊢ (𝑦 ∈ (V × V) ↔ 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
18	17	biimpri 228	. . . . 5 ⊢ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ → 𝑦 ∈ (V × V))
19		2nd1st 8019	. . . . 5 ⊢ (𝑦 ∈ (V × V) → ∪ ^◡{𝑦} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
20	16, 18, 19	3syl 18	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → ∪ ^◡{𝑦} = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
21		sneq 4601	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
22	21	cnveqd 5841	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ^◡{𝑥} = ^◡{𝑦})
23	22	unieqd 4886	. . . . . 6 ⊢ (𝑥 = 𝑦 → ∪ ^◡{𝑥} = ∪ ^◡{𝑦})
24	23, 13, 12	fvmpt3i 6975	. . . . 5 ⊢ (𝑦 ∈ 𝑅 → ((𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})‘𝑦) = ∪ ^◡{𝑦})
25	24	adantl 481	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → ((𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})‘𝑦) = ∪ ^◡{𝑦})
26	16	fveq2d 6864	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩))
27		ovtpos 8222	. . . . . . 7 ⊢ ((1^st ‘𝑦)tpos ( I ↾ ^◡𝑅)(2^nd ‘𝑦)) = ((2^nd ‘𝑦)( I ↾ ^◡𝑅)(1^st ‘𝑦))
28		df-ov 7392	. . . . . . 7 ⊢ ((1^st ‘𝑦)tpos ( I ↾ ^◡𝑅)(2^nd ‘𝑦)) = (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
29		df-ov 7392	. . . . . . 7 ⊢ ((2^nd ‘𝑦)( I ↾ ^◡𝑅)(1^st ‘𝑦)) = (( I ↾ ^◡𝑅)‘⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
30	27, 28, 29	3eqtr3i 2761	. . . . . 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 484	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝑦 ∈ 𝑅)
33	16, 32	eqeltrrd 2830	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ 𝑅)
34		fvex 6873	. . . . . . . 8 ⊢ (2^nd ‘𝑦) ∈ V
35		fvex 6873	. . . . . . . 8 ⊢ (1^st ‘𝑦) ∈ V
36	34, 35	opelcnv 5847	. . . . . . 7 ⊢ (⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ ∈ ^◡𝑅 ↔ ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ 𝑅)
37	36	biimpri 228	. . . . . 6 ⊢ (⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ 𝑅 → ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩ ∈ ^◡𝑅)
38		fvresi 7149	. . . . . 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 18	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (( I ↾ ^◡𝑅)‘⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
40	26, 31, 39	3eqtrd 2769	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
41	20, 25, 40	3eqtr4rd 2776	. . 3 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = ((𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})‘𝑦))
42	9, 15, 41	eqfnfvd 7008	. 2 ⊢ (Rel 𝑅 → tpos ( I ↾ ^◡𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}))
43	1, 42	eqtrd 2765	1 ⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3450 {csn 4591 ⟨cop 4597 ∪ cuni 4873 ↦ cmpt 5190 I cid 5534 × cxp 5638 ^◡ccnv 5639 ↾ cres 5642 Rel wrel 5645 Fn wfn 6508 ‘cfv 6513 (class class class)co 7389 1^st c1st 7968 2^nd c2nd 7969 tpos ctpos 8206

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-ov 7392 df-1st 7970 df-2nd 7971 df-tpos 8207

This theorem is referenced by: tposideq2 48865

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