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Theorem disjdsct 28651
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 5757) (Contributed by Thierry Arnoux, 28-Feb-2017.)
Hypotheses
Ref Expression
disjdsct.0 𝑥𝜑
disjdsct.1 𝑥𝐴
disjdsct.2 ((𝜑𝑥𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))
disjdsct.3 (𝜑Disj 𝑥𝐴 𝐵)
Assertion
Ref Expression
disjdsct (𝜑 → Fun (𝑥𝐴𝐵))
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.
StepHypRef Expression
1 disjdsct.3 . . . . . . . 8 (𝜑Disj 𝑥𝐴 𝐵)
2 disjdsct.1 . . . . . . . . 9 𝑥𝐴
32disjorsf 28563 . . . . . . . 8 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
41, 3sylib 206 . . . . . . 7 (𝜑 → ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
54r19.21bi 2820 . . . . . 6 ((𝜑𝑖𝐴) → ∀𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
65r19.21bi 2820 . . . . 5 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
7 simpr3 1061 . . . . . . . . 9 ((𝜑 ∧ (𝑖𝐴𝑗𝐴 ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)) → (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)
8 disjdsct.2 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))
9 eldifsni 4164 . . . . . . . . . . . . 13 (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
108, 9syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
1110sbimi 1836 . . . . . . . . . . 11 ([𝑖 / 𝑥](𝜑𝑥𝐴) → [𝑖 / 𝑥]𝐵 ≠ ∅)
12 sban 2291 . . . . . . . . . . . 12 ([𝑖 / 𝑥](𝜑𝑥𝐴) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥𝐴))
13 disjdsct.0 . . . . . . . . . . . . . 14 𝑥𝜑
1413sbf 2272 . . . . . . . . . . . . 13 ([𝑖 / 𝑥]𝜑𝜑)
152clelsb3f 28492 . . . . . . . . . . . . 13 ([𝑖 / 𝑥]𝑥𝐴𝑖𝐴)
1614, 15anbi12i 728 . . . . . . . . . . . 12 (([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥𝐴) ↔ (𝜑𝑖𝐴))
1712, 16bitri 262 . . . . . . . . . . 11 ([𝑖 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴))
18 sbsbc 3310 . . . . . . . . . . . 12 ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ [𝑖 / 𝑥]𝐵 ≠ ∅)
19 sbcne12 3841 . . . . . . . . . . . 12 ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ 𝑖 / 𝑥𝐵𝑖 / 𝑥∅)
20 csb0 3837 . . . . . . . . . . . . 13 𝑖 / 𝑥∅ = ∅
2120neeq2i 2751 . . . . . . . . . . . 12 (𝑖 / 𝑥𝐵𝑖 / 𝑥∅ ↔ 𝑖 / 𝑥𝐵 ≠ ∅)
2218, 19, 213bitri 284 . . . . . . . . . . 11 ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ 𝑖 / 𝑥𝐵 ≠ ∅)
2311, 17, 223imtr3i 278 . . . . . . . . . 10 ((𝜑𝑖𝐴) → 𝑖 / 𝑥𝐵 ≠ ∅)
24233ad2antr1 1218 . . . . . . . . 9 ((𝜑 ∧ (𝑖𝐴𝑗𝐴 ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)) → 𝑖 / 𝑥𝐵 ≠ ∅)
25 disj3 3876 . . . . . . . . . . . . 13 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ ↔ 𝑖 / 𝑥𝐵 = (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
2625biimpi 204 . . . . . . . . . . . 12 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ → 𝑖 / 𝑥𝐵 = (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
2726neeq1d 2745 . . . . . . . . . . 11 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ → (𝑖 / 𝑥𝐵 ≠ ∅ ↔ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) ≠ ∅))
2827biimpa 499 . . . . . . . . . 10 (((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ ∧ 𝑖 / 𝑥𝐵 ≠ ∅) → (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) ≠ ∅)
29 difn0 3800 . . . . . . . . . 10 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) ≠ ∅ → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
3028, 29syl 17 . . . . . . . . 9 (((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ ∧ 𝑖 / 𝑥𝐵 ≠ ∅) → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
317, 24, 30syl2anc 690 . . . . . . . 8 ((𝜑 ∧ (𝑖𝐴𝑗𝐴 ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)) → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
32313anassrs 1281 . . . . . . 7 ((((𝜑𝑖𝐴) ∧ 𝑗𝐴) ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅) → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
3332ex 448 . . . . . 6 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
3433orim2d 880 . . . . 5 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → ((𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅) → (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
356, 34mpd 15 . . . 4 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
3635ralrimiva 2853 . . 3 ((𝜑𝑖𝐴) → ∀𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
3736ralrimiva 2853 . 2 (𝜑 → ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
38 nfmpt1 4573 . . 3 𝑥(𝑥𝐴𝐵)
39 eqid 2514 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
4013, 2, 38, 39, 8funcnv4mpt 28641 . 2 (𝜑 → (Fun (𝑥𝐴𝐵) ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
4137, 40mpbird 245 1 (𝜑 → Fun (𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wa 382  w3a 1030   = wceq 1474  wnf 1698  [wsb 1830  wcel 1938  wnfc 2642  wne 2684  wral 2800  [wsbc 3306  csb 3403  cdif 3441  cin 3443  c0 3777  {csn 4028  Disj wdisj 4451  cmpt 4541  ccnv 4931  Fun wfun 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-disj 4452  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-fv 5697
This theorem is referenced by:  esumrnmpt  29237  measvunilem  29398
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