Theorem disjdsct 29480

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 5958) (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 29393	. . . . . . . 8 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
4	1, 3	sylib 208	. . . . . . 7 ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
5	4	r19.21bi 2932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
6	5	r19.21bi 2932	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) → (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
7		simpr3 1069	. . . . . . . . 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 4320	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
10	8, 9	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
11	10	sbimi 1886	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑖 / 𝑥]𝐵 ≠ ∅)
12		sban 2399	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥 ∈ 𝐴))
13		disjdsct.0	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝜑
14	13	sbf 2380	. . . . . . . . . . . . 13 ⊢ ([𝑖 / 𝑥]𝜑 ↔ 𝜑)
15	2	clelsb3f 2768	. . . . . . . . . . . . 13 ⊢ ([𝑖 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴)
16	14, 15	anbi12i 733	. . . . . . . . . . . 12 ⊢ (([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴))
17	12, 16	bitri 264	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑖 ∈ 𝐴))
18		sbsbc 3439	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ [𝑖 / 𝑥]𝐵 ≠ ∅)
19		sbcne12 3986	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑖 / 𝑥⦌∅)
20		csb0 3982	. . . . . . . . . . . . 13 ⊢ ⦋𝑖 / 𝑥⦌∅ = ∅
21	20	neeq2i 2859	. . . . . . . . . . . 12 ⊢ (⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑖 / 𝑥⦌∅ ↔ ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅)
22	18, 19, 21	3bitri 286	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅)
23	11, 17, 22	3imtr3i 280	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅)
24	23	3ad2antr1 1226	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ∧ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅)
25		disj3 4021	. . . . . . . . . . . . 13 ⊢ ((⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅ ↔ ⦋𝑖 / 𝑥⦌𝐵 = (⦋𝑖 / 𝑥⦌𝐵 ∖ ⦋𝑗 / 𝑥⦌𝐵))
26	25	biimpi 206	. . . . . . . . . . . 12 ⊢ ((⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅ → ⦋𝑖 / 𝑥⦌𝐵 = (⦋𝑖 / 𝑥⦌𝐵 ∖ ⦋𝑗 / 𝑥⦌𝐵))
27	26	neeq1d 2853	. . . . . . . . . . 11 ⊢ ((⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅ → (⦋𝑖 / 𝑥⦌𝐵 ≠ ∅ ↔ (⦋𝑖 / 𝑥⦌𝐵 ∖ ⦋𝑗 / 𝑥⦌𝐵) ≠ ∅))
28	27	biimpa 501	. . . . . . . . . 10 ⊢ (((⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅ ∧ ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅) → (⦋𝑖 / 𝑥⦌𝐵 ∖ ⦋𝑗 / 𝑥⦌𝐵) ≠ ∅)
29		difn0 3943	. . . . . . . . . 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 693	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ∧ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)
32	31	3anassrs 1290	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) ∧ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)
33	32	ex 450	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) → ((⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅ → ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))
34	33	orim2d 885	. . . . 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 2966	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))
37	36	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))
38		nfmpt1 4747	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
39		eqid 2622	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
40	13, 2, 38, 39, 8	funcnv4mpt 29470	. 2 ⊢ (𝜑 → (Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)))
41	37, 40	mpbird 247	1 ⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵))

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483  Ⅎwnf 1708  [wsb 1880   ∈ wcel 1990  Ⅎwnfc 2751   ≠ wne 2794  ∀wral 2912  [wsbc 3435  ⦋csb 3533   ∖ cdif 3571   ∩ cin 3573  ∅c0 3915  {csn 4177  Disj wdisj 4620   ↦ cmpt 4729  ^◡ccnv 5113  Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  esumrnmpt  30114  measvunilem  30275