Theorem disjdsct 32776

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 6567) (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 32650	. . . . . . . 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 3229	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
6	5	r19.21bi 3229	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) → (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
7		simpr3 1198	. . . . . . . . 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 4735	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
10	8, 9	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)
11	10	sbimi 2080	. . . . . . . . . . 11 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑖 / 𝑥]𝐵 ≠ ∅)
12		sban 2086	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥 ∈ 𝐴))
13		disjdsct.0	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝜑
14	13	sbf 2278	. . . . . . . . . . . . 13 ⊢ ([𝑖 / 𝑥]𝜑 ↔ 𝜑)
15	2	clelsb1fw 2902	. . . . . . . . . . . . 13 ⊢ ([𝑖 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴)
16	14, 15	anbi12i 629	. . . . . . . . . . . 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 3732	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ [𝑖 / 𝑥]𝐵 ≠ ∅)
19		sbcne12 4355	. . . . . . . . . . . 12 ⊢ ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑖 / 𝑥⦌∅)
20		csb0 4350	. . . . . . . . . . . . 13 ⊢ ⦋𝑖 / 𝑥⦌∅ = ∅
21	20	neeq2i 2997	. . . . . . . . . . . 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 1190	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ∧ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ∅)
25		disj3 4394	. . . . . . . . . . . . 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 2991	. . . . . . . . . . 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 4307	. . . . . . . . . 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 585	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ∧ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) → ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)
32	31	3anassrs 1362	. . . . . . 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 969	. . . . 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 3129	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))
37	36	ralrimiva 3129	. 2 ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵))
38		nfmpt1 5184	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
39		eqid 2736	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
40	13, 2, 38, 39, 8	funcnv4mpt 32741	. 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 848   ∧ w3a 1087   = wceq 1542  Ⅎwnf 1785  [wsb 2068   ∈ wcel 2114  Ⅎwnfc 2883   ≠ wne 2932  ∀wral 3051  [wsbc 3728  ⦋csb 3837   ∖ cdif 3886   ∩ cin 3888  ∅c0 4273  {csn 4567  Disj wdisj 5052   ↦ cmpt 5166  ^◡ccnv 5630  Fun wfun 6492

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 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-disj 5053  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-fv 6506

This theorem is referenced by:  esumrnmpt  34196  measvunilem  34356