Theorem disjdsct 32599

Description: A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 6569) (Contributed by Thierry Arnoux, 28-Feb-2017.)

Hypotheses

Ref	Expression
disjdsct.0	⊢ Ⅎ𝑥𝜑
disjdsct.1	⊢ Ⅎ𝑥𝐴
disjdsct.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))
disjdsct.3	⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)

Assertion

Ref	Expression
disjdsct	⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵))

Distinct variable group:   𝑥,𝑉

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem disjdsct

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjdsct.3	. . . . . . . 8 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
2		disjdsct.1	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
3	2	disjorsf 32482	. . . . . . . 8 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
4	1, 3	sylib 218	. . . . . . 7 ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
5	4	r19.21bi 3227	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
6	5	r19.21bi 3227	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) → (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
7		simpr3 1197	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ∧ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) → (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)
8		disjdsct.2	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))
9		eldifsni 4750	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
10	8, 9	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
11	10	sbimi 2075	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑖 / 𝑥]𝐵 ≠ ∅)
12		sban 2081	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥 ∈ 𝐴))
13		disjdsct.0	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝜑
14	13	sbf 2271	. . . . . . . . . . . . 13 ⊢ ([𝑖 / 𝑥]𝜑 ↔ 𝜑)
15	2	clelsb1fw 2895	. . . . . . . . . . . . 13 ⊢ ([𝑖 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴)
16	14, 15	anbi12i 628	. . . . . . . . . . . 12 ⊢ (([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴))
17	12, 16	bitri 275	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴))
18		sbsbc 3754	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ [𝑖 / 𝑥]𝐵 ≠ ∅)
19		sbcne12 4374	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑖 / 𝑥⦌∅)
20		csb0 4369	. . . . . . . . . . . . 13 ⊢ ⦋𝑖 / 𝑥⦌∅ = ∅
21	20	neeq2i 2990	. . . . . . . . . . . 12 ⊢ (⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑖 / 𝑥⦌∅ ↔ ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅)
22	18, 19, 21	3bitri 297	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅)
23	11, 17, 22	3imtr3i 291	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅)
24	23	3ad2antr1 1189	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ∧ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅)
25		disj3 4413	. . . . . . . . . . . . 13 ⊢ ((⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅ ↔ ⦋𝑖 / 𝑥⦌𝐵 = (⦋𝑖 / 𝑥⦌𝐵 ∖ ⦋𝑗 / 𝑥⦌𝐵))
26	25	biimpi 216	. . . . . . . . . . . 12 ⊢ ((⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅ → ⦋𝑖 / 𝑥⦌𝐵 = (⦋𝑖 / 𝑥⦌𝐵 ∖ ⦋𝑗 / 𝑥⦌𝐵))
27	26	neeq1d 2984	. . . . . . . . . . 11 ⊢ ((⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅ → (⦋𝑖 / 𝑥⦌𝐵 ≠ ∅ ↔ (⦋𝑖 / 𝑥⦌𝐵 ∖ ⦋𝑗 / 𝑥⦌𝐵) ≠ ∅))
28	27	biimpa 476	. . . . . . . . . 10 ⊢ (((⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅ ∧ ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅) → (⦋𝑖 / 𝑥⦌𝐵 ∖ ⦋𝑗 / 𝑥⦌𝐵) ≠ ∅)
29		difn0 4326	. . . . . . . . . 10 ⊢ ((⦋𝑖 / 𝑥⦌𝐵 ∖ ⦋𝑗 / 𝑥⦌𝐵) ≠ ∅ → ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)
30	28, 29	syl 17	. . . . . . . . 9 ⊢ (((⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅ ∧ ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)
31	7, 24, 30	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ∧ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)
32	31	3anassrs 1361	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) ∧ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)
33	32	ex 412	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) → ((⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅ → ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))
34	33	orim2d 968	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) → ((𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅) → (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)))
35	6, 34	mpd 15	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) → (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))
36	35	ralrimiva 3125	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))
37	36	ralrimiva 3125	. 2 ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))
38		nfmpt1 5201	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
39		eqid 2729	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
40	13, 2, 38, 39, 8	funcnv4mpt 32566	. 2 ⊢ (𝜑 → (Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)))
41	37, 40	mpbird 257	1 ⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540  Ⅎwnf 1783  [wsb 2065   ∈ wcel 2109  Ⅎwnfc 2876   ≠ wne 2925  ∀wral 3044  [wsbc 3750  ⦋csb 3859   ∖ cdif 3908   ∩ cin 3910  ∅c0 4292  {csn 4585  Disj wdisj 5069   ↦ cmpt 5183  ^◡ccnv 5630  Fun wfun 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-disj 5070  df-br 5103  df-opab 5165  df-mpt 5184  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 6452  df-fun 6501  df-fn 6502  df-fv 6507

This theorem is referenced by:  esumrnmpt  34015  measvunilem  34175