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Theorem disjdsct 31616
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 6570) (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 31498 . . . . . . . 8 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
41, 3sylib 217 . . . . . . 7 (𝜑 → ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
54r19.21bi 3234 . . . . . 6 ((𝜑𝑖𝐴) → ∀𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
65r19.21bi 3234 . . . . 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 4750 . . . . . . . . . . . . 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 2911 . . . . . . . . . . . . 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 3743 . . . . . . . . . . . 12 ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ [𝑖 / 𝑥]𝐵 ≠ ∅)
19 sbcne12 4372 . . . . . . . . . . . 12 ([𝑖 / 𝑥]𝐵 ≠ ∅ ↔ 𝑖 / 𝑥𝐵𝑖 / 𝑥∅)
20 csb0 4367 . . . . . . . . . . . . 13 𝑖 / 𝑥∅ = ∅
2120neeq2i 3009 . . . . . . . . . . . 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 4413 . . . . . . . . . . . . 13 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ ↔ 𝑖 / 𝑥𝐵 = (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
2625biimpi 215 . . . . . . . . . . . 12 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ → 𝑖 / 𝑥𝐵 = (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
2726neeq1d 3003 . . . . . . . . . . 11 ((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ → (𝑖 / 𝑥𝐵 ≠ ∅ ↔ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) ≠ ∅))
2827biimpa 477 . . . . . . . . . 10 (((𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅ ∧ 𝑖 / 𝑥𝐵 ≠ ∅) → (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) ≠ ∅)
29 difn0 4324 . . . . . . . . . 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 3143 . . 3 ((𝜑𝑖𝐴) → ∀𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
3736ralrimiva 3143 . 2 (𝜑 → ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵))
38 nfmpt1 5213 . . 3 𝑥(𝑥𝐴𝐵)
39 eqid 2736 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
4013, 2, 38, 39, 8funcnv4mpt 31585 . 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 2887  wne 2943  wral 3064  [wsbc 3739  csb 3855  cdif 3907  cin 3909  c0 4282  {csn 4586  Disj wdisj 5070  cmpt 5188  ccnv 5632  Fun wfun 6490
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504
This theorem is referenced by:  esumrnmpt  32651  measvunilem  32811
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