Theorem trsp2cyc 33205

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 768	. . . . . . 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 5101	. . . . . . . 8 ⊢ (𝑦 = 𝑝 → (𝑦 ≈ 2o ↔ 𝑝 ≈ 2o))
3	2	elrab 3646	. . . . . . 7 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↔ (𝑝 ∈ 𝒫 𝐷 ∧ 𝑝 ≈ 2o))
4	1, 3	sylib 218	. . . . . 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 495	. . . . 5 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → 𝑝 ≈ 2o)
6		en2 9180	. . . . 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 494	. . . . . . . . . 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 4563	. . . . . . . . 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 480	. . . . . . . 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 3444	. . . . . . . . . 10 ⊢ 𝑖 ∈ V
12	11	prid1 4719	. . . . . . . . 9 ⊢ 𝑖 ∈ {𝑖, 𝑗}
13		simpr 484	. . . . . . . . 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 2843	. . . . . . . 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 3934	. . . . . . 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 3444	. . . . . . . . . 10 ⊢ 𝑗 ∈ V
17	16	prid2 4720	. . . . . . . . 9 ⊢ 𝑗 ∈ {𝑖, 𝑗}
18	17, 13	eleqtrrid 2843	. . . . . . . 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 3934	. . . . . . 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 480	. . . . . . . . . 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 5122	. . . . . . . . 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 9915	. . . . . . . . . 10 ⊢ ((𝑖 ∈ 𝐷 ∧ 𝑗 ∈ 𝐷) → ({𝑖, 𝑗} ≈ 2o ↔ 𝑖 ≠ 𝑗))
23	22	biimpa 476	. . . . . . . . 9 ⊢ (((𝑖 ∈ 𝐷 ∧ 𝑗 ∈ 𝐷) ∧ {𝑖, 𝑗} ≈ 2o) → 𝑖 ≠ 𝑗)
24	15, 19, 21, 23	syl21anc 837	. . . . . . . 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 768	. . . . . . . . . 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 782	. . . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
28	27	pmtrval 19380	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑝 ⊆ 𝐷 ∧ 𝑝 ≈ 2o) → ((pmTrsp‘𝐷)‘𝑝) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
29	26, 10, 20, 28	syl3anc 1373	. . . . . . . . . 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 6838	. . . . . . . . . 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 2777	. . . . . . . . 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 33201	. . . . . . . . 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 2774	. . . . . . . 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 511	. . . . . . 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 514	. . . . . 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 412	. . . . 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 1920	. . . 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 3173	. . 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 234	. 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 484	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → 𝑃 ∈ 𝑇)
43		trsp2cyc.t	. . . . 5 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
44	27	pmtrfval 19379	. . . . . . 7 ⊢ (𝐷 ∈ 𝑉 → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
45	44	adantr 480	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
46	45	rneqd 5887	. . . . 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 2783	. . . 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 2838	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → 𝑃 ∈ ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
49		eqid 2736	. . . . 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 5907	. . . 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 481	. . 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 232	. 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 3144	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → ∃𝑖 ∈ 𝐷 ∃𝑗 ∈ 𝐷 (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060  {crab 3399   ∖ cdif 3898   ⊆ wss 3901  ifcif 4479  𝒫 cpw 4554  {csn 4580  {cpr 4582  ∪ cuni 4863   class class class wbr 5098   ↦ cmpt 5179  ran crn 5625  ‘cfv 6492  2_oc2o 8391   ≈ cen 8880  ⟨“cs2 14764  pmTrspcpmtr 19370  toCycctocyc 33188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-n0 12402  df-xnn0 12475  df-z 12489  df-uz 12752  df-rp 12906  df-fz 13424  df-fzo 13571  df-fl 13712  df-mod 13790  df-hash 14254  df-word 14437  df-concat 14494  df-s1 14520  df-substr 14565  df-pfx 14595  df-csh 14712  df-s2 14771  df-pmtr 19371  df-tocyc 33189

This theorem is referenced by:  cyc3genpm  33234