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Theorem disjdsct 31962
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 6617) (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 31849 . . . . . . . 8 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
41, 3sylib 217 . . . . . . 7 (𝜑 → ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
54r19.21bi 3248 . . . . . 6 ((𝜑𝑖𝐴) → ∀𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
65r19.21bi 3248 . . . . 5 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
7 simpr3 1196 . . . . . . . . 9 ((𝜑 ∧ (𝑖𝐴𝑗𝐴 ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)) → (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)
8 disjdsct.2 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))
9 eldifsni 4793 . . . . . . . . . . . . 13 (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
108, 9syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
1110sbimi 2077 . . . . . . . . . . 11 ([𝑖 / 𝑥](𝜑𝑥𝐴) → [𝑖 / 𝑥]𝐵 ≠ ∅)
12 sban 2083 . . . . . . . . . . . 12 ([𝑖 / 𝑥](𝜑𝑥𝐴) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥𝐴))
13 disjdsct.0 . . . . . . . . . . . . . 14 𝑥𝜑
1413sbf 2262 . . . . . . . . . . . . 13 ([𝑖 / 𝑥]𝜑𝜑)
152clelsb1fw 2907 . . . . . . . . . . . . 13 ([𝑖 / 𝑥]𝑥𝐴𝑖𝐴)
1614, 15anbi12i 627 . . . . . . . . . . . 12 (([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥𝐴) ↔ (𝜑𝑖𝐴))
1712, 16bitri 274 . . . . . . . . . . 11 ([𝑖 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴))
18 sbsbc 3781 . . . . . . . . . . . 12 ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ [𝑖 / 𝑥]𝐵 ≠ ∅)
19 sbcne12 4412 . . . . . . . . . . . 12 ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ 𝑖 / 𝑥𝐵𝑖 / 𝑥∅)
20 csb0 4407 . . . . . . . . . . . . 13 𝑖 / 𝑥∅ = ∅
2120neeq2i 3006 . . . . . . . . . . . 12 (𝑖 / 𝑥𝐵𝑖 / 𝑥∅ ↔ 𝑖 / 𝑥𝐵 ≠ ∅)
2218, 19, 213bitri 296 . . . . . . . . . . 11 ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ 𝑖 / 𝑥𝐵 ≠ ∅)
2311, 17, 223imtr3i 290 . . . . . . . . . 10 ((𝜑𝑖𝐴) → 𝑖 / 𝑥𝐵 ≠ ∅)
24233ad2antr1 1188 . . . . . . . . 9 ((𝜑 ∧ (𝑖𝐴𝑗𝐴 ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)) → 𝑖 / 𝑥𝐵 ≠ ∅)
25 disj3 4453 . . . . . . . . . . . . 13 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ ↔ 𝑖 / 𝑥𝐵 = (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
2625biimpi 215 . . . . . . . . . . . 12 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ → 𝑖 / 𝑥𝐵 = (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
2726neeq1d 3000 . . . . . . . . . . 11 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ → (𝑖 / 𝑥𝐵 ≠ ∅ ↔ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) ≠ ∅))
2827biimpa 477 . . . . . . . . . 10 (((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ ∧ 𝑖 / 𝑥𝐵 ≠ ∅) → (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) ≠ ∅)
29 difn0 4364 . . . . . . . . . 10 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) ≠ ∅ → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
3028, 29syl 17 . . . . . . . . 9 (((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ ∧ 𝑖 / 𝑥𝐵 ≠ ∅) → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
317, 24, 30syl2anc 584 . . . . . . . 8 ((𝜑 ∧ (𝑖𝐴𝑗𝐴 ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)) → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
32313anassrs 1360 . . . . . . 7 ((((𝜑𝑖𝐴) ∧ 𝑗𝐴) ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅) → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
3332ex 413 . . . . . 6 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
3433orim2d 965 . . . . 5 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → ((𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅) → (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
356, 34mpd 15 . . . 4 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
3635ralrimiva 3146 . . 3 ((𝜑𝑖𝐴) → ∀𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
3736ralrimiva 3146 . 2 (𝜑 → ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
38 nfmpt1 5256 . . 3 𝑥(𝑥𝐴𝐵)
39 eqid 2732 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
4013, 2, 38, 39, 8funcnv4mpt 31932 . 2 (𝜑 → (Fun (𝑥𝐴𝐵) ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
4137, 40mpbird 256 1 (𝜑 → Fun (𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845  w3a 1087   = wceq 1541  wnf 1785  [wsb 2067  wcel 2106  wnfc 2883  wne 2940  wral 3061  [wsbc 3777  csb 3893  cdif 3945  cin 3947  c0 4322  {csn 4628  Disj wdisj 5113  cmpt 5231  ccnv 5675  Fun wfun 6537
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by:  esumrnmpt  33119  measvunilem  33279
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