tposideq - Mathbox for Zhi Wang

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Theorem tposideq 49363

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 49357	. 2 ⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = tpos ( I ↾ ^◡𝑅))
2		relcnv 6069	. . . . 5 ⊢ Rel ^◡𝑅
3		fnresi 6627	. . . . 5 ⊢ ( I ↾ ^◡𝑅) Fn ^◡𝑅
4		tposfn2 8198	. . . . 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 6153	. . . . . 6 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)
7	6	biimpi 216	. . . . 5 ⊢ (Rel 𝑅 → ^◡^◡𝑅 = 𝑅)
8	7	fneq2d 6592	. . . 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 5377	. . . . . . 7 ⊢ {𝑥} ∈ V
11	10	cnvex 7876	. . . . . 6 ⊢ ^◡{𝑥} ∈ V
12	11	uniex 7695	. . . . 5 ⊢ ∪ ^◡{𝑥} ∈ V
13		eqid 2736	. . . . 5 ⊢ (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})
14	12, 13	fnmpti 6641	. . . 4 ⊢ (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) Fn 𝑅
15	14	a1i 11	. . 3 ⊢ (Rel 𝑅 → (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}) Fn 𝑅)
16		1st2nd 7992	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
17		1st2ndb 7982	. . . . . 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 7991	. . . . 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 4577	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
22	21	cnveqd 5830	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ^◡{𝑥} = ^◡{𝑦})
23	22	unieqd 4863	. . . . . 6 ⊢ (𝑥 = 𝑦 → ∪ ^◡{𝑥} = ∪ ^◡{𝑦})
24	23, 13, 12	fvmpt3i 6953	. . . . 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 6844	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩))
27		ovtpos 8191	. . . . . . 7 ⊢ ((1^st ‘𝑦)tpos ( I ↾ ^◡𝑅)(2^nd ‘𝑦)) = ((2^nd ‘𝑦)( I ↾ ^◡𝑅)(1^st ‘𝑦))
28		df-ov 7370	. . . . . . 7 ⊢ ((1^st ‘𝑦)tpos ( I ↾ ^◡𝑅)(2^nd ‘𝑦)) = (tpos ( I ↾ ^◡𝑅)‘⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
29		df-ov 7370	. . . . . . 7 ⊢ ((2^nd ‘𝑦)( I ↾ ^◡𝑅)(1^st ‘𝑦)) = (( I ↾ ^◡𝑅)‘⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
30	27, 28, 29	3eqtr3i 2767	. . . . . 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 2837	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ 𝑅)
34		fvex 6853	. . . . . . . 8 ⊢ (2^nd ‘𝑦) ∈ V
35		fvex 6853	. . . . . . . 8 ⊢ (1^st ‘𝑦) ∈ V
36	34, 35	opelcnv 5836	. . . . . . 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 7128	. . . . . 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 2775	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = ⟨(2^nd ‘𝑦), (1^st ‘𝑦)⟩)
41	20, 25, 40	3eqtr4rd 2782	. . 3 ⊢ ((Rel 𝑅 ∧ 𝑦 ∈ 𝑅) → (tpos ( I ↾ ^◡𝑅)‘𝑦) = ((𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥})‘𝑦))
42	9, 15, 41	eqfnfvd 6986	. 2 ⊢ (Rel 𝑅 → tpos ( I ↾ ^◡𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}))
43	1, 42	eqtrd 2771	1 ⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ^◡{𝑥}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3429 {csn 4567 ⟨cop 4573 ∪ cuni 4850 ↦ cmpt 5166 I cid 5525 × cxp 5629 ^◡ccnv 5630 ↾ cres 5633 Rel wrel 5636 Fn wfn 6493 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 tpos ctpos 8175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-1st 7942 df-2nd 7943 df-tpos 8176

This theorem is referenced by: tposideq2 49364

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