Theorem trsp2cyc 31390

Description: Exhibit the word a transposition corresponds to, as a cycle. (Contributed by Thierry Arnoux, 25-Sep-2023.)

Hypotheses

Ref	Expression
trsp2cyc.t	⊢ 𝑇 = ran (pmTrsp‘𝐷)
trsp2cyc.c	⊢ 𝐶 = (toCyc‘𝐷)

Assertion

Ref	Expression
trsp2cyc	⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → ∃𝑖 ∈ 𝐷 ∃𝑗 ∈ 𝐷 (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))

Distinct variable groups:   𝐷,𝑖,𝑗   𝑃,𝑖,𝑗   𝑇,𝑖,𝑗   𝑖,𝑉,𝑗

Allowed substitution hints:   𝐶(𝑖,𝑗)

Proof of Theorem trsp2cyc

Dummy variables 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 766	. . . . . . 7 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o})
2		breq1 5077	. . . . . . . 8 ⊢ (𝑦 = 𝑝 → (𝑦 ≈ 2o ↔ 𝑝 ≈ 2o))
3	2	elrab 3624	. . . . . . 7 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↔ (𝑝 ∈ 𝒫 𝐷 ∧ 𝑝 ≈ 2o))
4	1, 3	sylib 217	. . . . . 6 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → (𝑝 ∈ 𝒫 𝐷 ∧ 𝑝 ≈ 2o))
5	4	simprd 496	. . . . 5 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → 𝑝 ≈ 2o)
6		en2 9053	. . . . 5 ⊢ (𝑝 ≈ 2o → ∃𝑖∃𝑗 𝑝 = {𝑖, 𝑗})
7	5, 6	syl 17	. . . 4 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → ∃𝑖∃𝑗 𝑝 = {𝑖, 𝑗})
8	4	simpld 495	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → 𝑝 ∈ 𝒫 𝐷)
9	8	elpwid 4544	. . . . . . . . 9 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → 𝑝 ⊆ 𝐷)
10	9	adantr 481	. . . . . . . 8 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝 ⊆ 𝐷)
11		vex 3436	. . . . . . . . . 10 ⊢ 𝑖 ∈ V
12	11	prid1 4698	. . . . . . . . 9 ⊢ 𝑖 ∈ {𝑖, 𝑗}
13		simpr 485	. . . . . . . . 9 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝 = {𝑖, 𝑗})
14	12, 13	eleqtrrid 2846	. . . . . . . 8 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑖 ∈ 𝑝)
15	10, 14	sseldd 3922	. . . . . . 7 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑖 ∈ 𝐷)
16		vex 3436	. . . . . . . . . 10 ⊢ 𝑗 ∈ V
17	16	prid2 4699	. . . . . . . . 9 ⊢ 𝑗 ∈ {𝑖, 𝑗}
18	17, 13	eleqtrrid 2846	. . . . . . . 8 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑗 ∈ 𝑝)
19	10, 18	sseldd 3922	. . . . . . 7 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑗 ∈ 𝐷)
20	5	adantr 481	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝 ≈ 2o)
21	13, 20	eqbrtrrd 5098	. . . . . . . . 9 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → {𝑖, 𝑗} ≈ 2o)
22		pr2ne 9761	. . . . . . . . . 10 ⊢ ((𝑖 ∈ 𝐷 ∧ 𝑗 ∈ 𝐷) → ({𝑖, 𝑗} ≈ 2o ↔ 𝑖 ≠ 𝑗))
23	22	biimpa 477	. . . . . . . . 9 ⊢ (((𝑖 ∈ 𝐷 ∧ 𝑗 ∈ 𝐷) ∧ {𝑖, 𝑗} ≈ 2o) → 𝑖 ≠ 𝑗)
24	15, 19, 21, 23	syl21anc 835	. . . . . . . 8 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑖 ≠ 𝑗)
25		simplr 766	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
26		simp-4l 780	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝐷 ∈ 𝑉)
27		eqid 2738	. . . . . . . . . . . 12 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
28	27	pmtrval 19059	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑝 ⊆ 𝐷 ∧ 𝑝 ≈ 2o) → ((pmTrsp‘𝐷)‘𝑝) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
29	26, 10, 20, 28	syl3anc 1370	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → ((pmTrsp‘𝐷)‘𝑝) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
30	13	fveq2d 6778	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → ((pmTrsp‘𝐷)‘𝑝) = ((pmTrsp‘𝐷)‘{𝑖, 𝑗}))
31	25, 29, 30	3eqtr2d 2784	. . . . . . . . 9 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑃 = ((pmTrsp‘𝐷)‘{𝑖, 𝑗}))
32		trsp2cyc.c	. . . . . . . . . 10 ⊢ 𝐶 = (toCyc‘𝐷)
33	32, 26, 15, 19, 24, 27	cycpm2tr 31386	. . . . . . . . 9 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → (𝐶‘⟨“𝑖𝑗”⟩) = ((pmTrsp‘𝐷)‘{𝑖, 𝑗}))
34	31, 33	eqtr4d 2781	. . . . . . . 8 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩))
35	24, 34	jca 512	. . . . . . 7 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))
36	15, 19, 35	jca31 515	. . . . . 6 ⊢ (((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → ((𝑖 ∈ 𝐷 ∧ 𝑗 ∈ 𝐷) ∧ (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩))))
37	36	ex 413	. . . . 5 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → (𝑝 = {𝑖, 𝑗} → ((𝑖 ∈ 𝐷 ∧ 𝑗 ∈ 𝐷) ∧ (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))))
38	37	2eximdv 1922	. . . 4 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → (∃𝑖∃𝑗 𝑝 = {𝑖, 𝑗} → ∃𝑖∃𝑗((𝑖 ∈ 𝐷 ∧ 𝑗 ∈ 𝐷) ∧ (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))))
39	7, 38	mpd 15	. . 3 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → ∃𝑖∃𝑗((𝑖 ∈ 𝐷 ∧ 𝑗 ∈ 𝐷) ∧ (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩))))
40		r2ex 3232	. . 3 ⊢ (∃𝑖 ∈ 𝐷 ∃𝑗 ∈ 𝐷 (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)) ↔ ∃𝑖∃𝑗((𝑖 ∈ 𝐷 ∧ 𝑗 ∈ 𝐷) ∧ (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩))))
41	39, 40	sylibr 233	. 2 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → ∃𝑖 ∈ 𝐷 ∃𝑗 ∈ 𝐷 (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))
42		simpr 485	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → 𝑃 ∈ 𝑇)
43		trsp2cyc.t	. . . . 5 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
44	27	pmtrfval 19058	. . . . . . 7 ⊢ (𝐷 ∈ 𝑉 → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
45	44	adantr 481	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
46	45	rneqd 5847	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → ran (pmTrsp‘𝐷) = ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
47	43, 46	eqtrid 2790	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → 𝑇 = ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
48	42, 47	eleqtrd 2841	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → 𝑃 ∈ ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
49		eqid 2738	. . . . 5 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
50	49	elrnmpt 5865	. . . 4 ⊢ (𝑃 ∈ 𝑇 → (𝑃 ∈ ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ↔ ∃𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
51	50	adantl 482	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → (𝑃 ∈ ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ↔ ∃𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
52	48, 51	mpbid 231	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → ∃𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
53	41, 52	r19.29a 3218	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → ∃𝑖 ∈ 𝐷 ∃𝑗 ∈ 𝐷 (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065  {crab 3068   ∖ cdif 3884   ⊆ wss 3887  ifcif 4459  𝒫 cpw 4533  {csn 4561  {cpr 4563  ∪ cuni 4839   class class class wbr 5074   ↦ cmpt 5157  ran crn 5590  ‘cfv 6433  2_oc2o 8291   ≈ cen 8730  ⟨“cs2 14554  pmTrspcpmtr 19049  toCycctocyc 31373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-hash 14045  df-word 14218  df-concat 14274  df-s1 14301  df-substr 14354  df-pfx 14384  df-csh 14502  df-s2 14561  df-pmtr 19050  df-tocyc 31374

This theorem is referenced by:  cyc3genpm  31419