tposideq - Mathbox for Zhi Wang

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Theorem tposideq 48771

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 48765	. 2 ⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = tpos ( I ↾ ^◡𝑅))
2		relcnv 6102	. . . . 5 ⊢ Rel ^◡𝑅
3		fnresi 6677	. . . . 5 ⊢ ( I ↾ ^◡𝑅) Fn ^◡𝑅
4		tposfn2 8255	. . . . 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 6189	. . . . . 6 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
7	6	biimpi 216	. . . . 5 ⊢ (Rel 𝑅 → ^◡^◡𝑅 = 𝑅)
8	7	fneq2d 6642	. . . 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 5414	. . . . . . 7 ⊢ {𝑥} ∈ V
11	10	cnvex 7929	. . . . . 6 ⊢ ^◡{𝑥} ∈ V
12	11	uniex 7743	. . . . 5 ⊢ ∪ ^◡{𝑥} ∈ V
13		eqid 2734	. . . . 5 ⊢ (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})
14	12, 13	fnmpti 6691	. . . 4 ⊢ (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) Fn 𝑅
15	14	a1i 11	. . 3 ⊢ (Rel 𝑅 → (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) Fn 𝑅)
16		1st2nd 8046	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
17		1st2ndb 8036	. . . . . 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 8045	. . . . 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 4616	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
22	21	cnveqd 5866	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ^◡{𝑥} = ^◡{𝑦})
23	22	unieqd 4900	. . . . . 6 ⊢ (𝑥 = 𝑦 → ∪ ^◡{𝑥} = ∪ ^◡{𝑦})
24	23, 13, 12	fvmpt3i 7001	. . . . 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 6890	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩))
27		ovtpos 8248	. . . . . . 7 ⊢ ((1^st ‘𝑦)tpos ( I ↾ ^◡𝑅)(2^nd ‘𝑦)) = ((2^nd ‘𝑦)( I ↾ ^◡𝑅)(1^st ‘𝑦))
28		df-ov 7416	. . . . . . 7 ⊢ ((1^st ‘𝑦)tpos ( I ↾ ^◡𝑅)(2^nd ‘𝑦)) = (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
29		df-ov 7416	. . . . . . 7 ⊢ ((2^nd ‘𝑦)( I ↾ ^◡𝑅)(1^st ‘𝑦)) = (( I ↾ ^◡𝑅)‘⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
30	27, 28, 29	3eqtr3i 2765	. . . . . 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 2834	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ 𝑅)
34		fvex 6899	. . . . . . . 8 ⊢ (2^nd ‘𝑦) ∈ V
35		fvex 6899	. . . . . . . 8 ⊢ (1^st ‘𝑦) ∈ V
36	34, 35	opelcnv 5872	. . . . . . 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 7175	. . . . . 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 2773	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
41	20, 25, 40	3eqtr4rd 2780	. . 3 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = ((𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})‘𝑦))
42	9, 15, 41	eqfnfvd 7034	. 2 ⊢ (Rel 𝑅 → tpos ( I ↾ ^◡𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}))
43	1, 42	eqtrd 2769	1 ⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3463 {csn 4606 ⟨cop 4612 ∪ cuni 4887 ↦ cmpt 5205 I cid 5557 × cxp 5663 ^◡ccnv 5664 ↾ cres 5667 Rel wrel 5670 Fn wfn 6536 ‘cfv 6541 (class class class)co 7413 1^st c1st 7994 2^nd c2nd 7995 tpos ctpos 8232

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7416 df-1st 7996 df-2nd 7997 df-tpos 8233

This theorem is referenced by: tposideq2 48772

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