Theorem disjdsct 32714

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 6647) (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 32602	. . . . . . . 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 3257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
6	5	r19.21bi 3257	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) → (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
7		simpr3 1196	. . . . . . . . 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 4815	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
10	8, 9	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
11	10	sbimi 2074	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑖 / 𝑥]𝐵 ≠ ∅)
12		sban 2080	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥 ∈ 𝐴))
13		disjdsct.0	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝜑
14	13	sbf 2272	. . . . . . . . . . . . 13 ⊢ ([𝑖 / 𝑥]𝜑 ↔ 𝜑)
15	2	clelsb1fw 2912	. . . . . . . . . . . . 13 ⊢ ([𝑖 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴)
16	14, 15	anbi12i 627	. . . . . . . . . . . 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 3808	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ [𝑖 / 𝑥]𝐵 ≠ ∅)
19		sbcne12 4438	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑖 / 𝑥⦌∅)
20		csb0 4433	. . . . . . . . . . . . 13 ⊢ ⦋𝑖 / 𝑥⦌∅ = ∅
21	20	neeq2i 3012	. . . . . . . . . . . 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 1188	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ∧ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅)
25		disj3 4477	. . . . . . . . . . . . 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 3006	. . . . . . . . . . 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 4390	. . . . . . . . . 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 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ∧ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)
32	31	3anassrs 1360	. . . . . . 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 967	. . . . 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 3152	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))
37	36	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))
38		nfmpt1 5274	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
39		eqid 2740	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
40	13, 2, 38, 39, 8	funcnv4mpt 32687	. 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 846   ∧ w3a 1087   = wceq 1537  Ⅎwnf 1781  [wsb 2064   ∈ wcel 2108  Ⅎwnfc 2893   ≠ wne 2946  ∀wral 3067  [wsbc 3804  ⦋csb 3921   ∖ cdif 3973   ∩ cin 3975  ∅c0 4352  {csn 4648  Disj wdisj 5133   ↦ cmpt 5249  ^◡ccnv 5699  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  esumrnmpt  34016  measvunilem  34176