Theorem trsp2cyc 33217

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 769	. . . . . . 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 5103	. . . . . . . 8 ⊢ (𝑦 = 𝑝 → (𝑦 ≈ 2o ↔ 𝑝 ≈ 2o))
3	2	elrab 3648	. . . . . . 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 9192	. . . . 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 4565	. . . . . . . . 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 3446	. . . . . . . . . 10 ⊢ 𝑖 ∈ V
12	11	prid1 4721	. . . . . . . . 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 2844	. . . . . . . 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 3936	. . . . . . 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 3446	. . . . . . . . . 10 ⊢ 𝑗 ∈ V
17	16	prid2 4722	. . . . . . . . 9 ⊢ 𝑗 ∈ {𝑖, 𝑗}
18	17, 13	eleqtrrid 2844	. . . . . . . 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 3936	. . . . . . 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 5124	. . . . . . . . 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 9927	. . . . . . . . . 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 838	. . . . . . . 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 769	. . . . . . . . . 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 783	. . . . . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
28	27	pmtrval 19392	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑝 ⊆ 𝐷 ∧ 𝑝 ≈ 2o) → ((pmTrsp‘𝐷)‘𝑝) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
29	26, 10, 20, 28	syl3anc 1374	. . . . . . . . . 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 6846	. . . . . . . . . 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 2778	. . . . . . . . 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 33213	. . . . . . . . 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 2775	. . . . . . . 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 1921	. . . 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 3175	. . 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 19391	. . . . . . 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 5895	. . . . 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 2784	. . . 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 2839	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → 𝑃 ∈ ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
49		eqid 2737	. . . . 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 5915	. . . 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 3146	1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → ∃𝑖 ∈ 𝐷 ∃𝑗 ∈ 𝐷 (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  {crab 3401   ∖ cdif 3900   ⊆ wss 3903  ifcif 4481  𝒫 cpw 4556  {csn 4582  {cpr 4584  ∪ cuni 4865   class class class wbr 5100   ↦ cmpt 5181  ran crn 5633  ‘cfv 6500  2_oc2o 8401   ≈ cen 8892  ⟨“cs2 14776  pmTrspcpmtr 19382  toCycctocyc 33200

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-n0 12414  df-xnn0 12487  df-z 12501  df-uz 12764  df-rp 12918  df-fz 13436  df-fzo 13583  df-fl 13724  df-mod 13802  df-hash 14266  df-word 14449  df-concat 14506  df-s1 14532  df-substr 14577  df-pfx 14607  df-csh 14724  df-s2 14783  df-pmtr 19383  df-tocyc 33201

This theorem is referenced by:  cyc3genpm  33246