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Theorem disjdsct 29608
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 5996) (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 29519 . . . . . . . 8 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
41, 3sylib 208 . . . . . . 7 (𝜑 → ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
54r19.21bi 2961 . . . . . 6 ((𝜑𝑖𝐴) → ∀𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
65r19.21bi 2961 . . . . 5 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
7 simpr3 1089 . . . . . . . . 9 ((𝜑 ∧ (𝑖𝐴𝑗𝐴 ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)) → (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)
8 disjdsct.2 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))
9 eldifsni 4353 . . . . . . . . . . . . 13 (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
108, 9syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
1110sbimi 1943 . . . . . . . . . . 11 ([𝑖 / 𝑥](𝜑𝑥𝐴) → [𝑖 / 𝑥]𝐵 ≠ ∅)
12 sban 2427 . . . . . . . . . . . 12 ([𝑖 / 𝑥](𝜑𝑥𝐴) ↔ ([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥𝐴))
13 disjdsct.0 . . . . . . . . . . . . . 14 𝑥𝜑
1413sbf 2408 . . . . . . . . . . . . 13 ([𝑖 / 𝑥]𝜑𝜑)
152clelsb3f 2797 . . . . . . . . . . . . 13 ([𝑖 / 𝑥]𝑥𝐴𝑖𝐴)
1614, 15anbi12i 733 . . . . . . . . . . . 12 (([𝑖 / 𝑥]𝜑 ∧ [𝑖 / 𝑥]𝑥𝐴) ↔ (𝜑𝑖𝐴))
1712, 16bitri 264 . . . . . . . . . . 11 ([𝑖 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴))
18 sbsbc 3472 . . . . . . . . . . . 12 ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ [𝑖 / 𝑥]𝐵 ≠ ∅)
19 sbcne12 4019 . . . . . . . . . . . 12 ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ 𝑖 / 𝑥𝐵𝑖 / 𝑥∅)
20 csb0 4015 . . . . . . . . . . . . 13 𝑖 / 𝑥∅ = ∅
2120neeq2i 2888 . . . . . . . . . . . 12 (𝑖 / 𝑥𝐵𝑖 / 𝑥∅ ↔ 𝑖 / 𝑥𝐵 ≠ ∅)
2218, 19, 213bitri 286 . . . . . . . . . . 11 ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ 𝑖 / 𝑥𝐵 ≠ ∅)
2311, 17, 223imtr3i 280 . . . . . . . . . 10 ((𝜑𝑖𝐴) → 𝑖 / 𝑥𝐵 ≠ ∅)
24233ad2antr1 1246 . . . . . . . . 9 ((𝜑 ∧ (𝑖𝐴𝑗𝐴 ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)) → 𝑖 / 𝑥𝐵 ≠ ∅)
25 disj3 4054 . . . . . . . . . . . . 13 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ ↔ 𝑖 / 𝑥𝐵 = (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
2625biimpi 206 . . . . . . . . . . . 12 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ → 𝑖 / 𝑥𝐵 = (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
2726neeq1d 2882 . . . . . . . . . . 11 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ → (𝑖 / 𝑥𝐵 ≠ ∅ ↔ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) ≠ ∅))
2827biimpa 500 . . . . . . . . . 10 (((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ ∧ 𝑖 / 𝑥𝐵 ≠ ∅) → (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) ≠ ∅)
29 difn0 3976 . . . . . . . . . 10 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) ≠ ∅ → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
3028, 29syl 17 . . . . . . . . 9 (((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ ∧ 𝑖 / 𝑥𝐵 ≠ ∅) → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
317, 24, 30syl2anc 694 . . . . . . . 8 ((𝜑 ∧ (𝑖𝐴𝑗𝐴 ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅)) → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
32313anassrs 1313 . . . . . . 7 ((((𝜑𝑖𝐴) ∧ 𝑗𝐴) ∧ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅) → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)
3332ex 449 . . . . . 6 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ → 𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
3433orim2d 903 . . . . 5 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → ((𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅) → (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
356, 34mpd 15 . . . 4 (((𝜑𝑖𝐴) ∧ 𝑗𝐴) → (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
3635ralrimiva 2995 . . 3 ((𝜑𝑖𝐴) → ∀𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
3736ralrimiva 2995 . 2 (𝜑 → ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
38 nfmpt1 4780 . . 3 𝑥(𝑥𝐴𝐵)
39 eqid 2651 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
4013, 2, 38, 39, 8funcnv4mpt 29598 . 2 (𝜑 → (Fun (𝑥𝐴𝐵) ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
4137, 40mpbird 247 1 (𝜑 → Fun (𝑥𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1054   = wceq 1523  wnf 1748  [wsb 1937  wcel 2030  wnfc 2780  wne 2823  wral 2941  [wsbc 3468  csb 3566  cdif 3604  cin 3606  c0 3948  {csn 4210  Disj wdisj 4652  cmpt 4762  ccnv 5142  Fun wfun 5920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  esumrnmpt  30242  measvunilem  30403
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