Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funcnv4mpt Structured version   Visualization version   GIF version

Theorem funcnv4mpt 31006
Description: Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)
Hypotheses
Ref Expression
funcnvmpt.0 𝑥𝜑
funcnvmpt.1 𝑥𝐴
funcnvmpt.2 𝑥𝐹
funcnvmpt.3 𝐹 = (𝑥𝐴𝐵)
funcnvmpt.4 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
funcnv4mpt (𝜑 → (Fun 𝐹 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
Distinct variable groups:   𝑖,𝑗,𝑥   𝐴,𝑖,𝑗   𝐵,𝑖,𝑗   𝑖,𝐹   𝑥,𝑉   𝜑,𝑖,𝑗
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem funcnv4mpt
StepHypRef Expression
1 nfv 1917 . 2 𝑖𝜑
2 nfcv 2907 . 2 𝑖𝐴
3 nfcv 2907 . 2 𝑖𝐹
4 funcnvmpt.3 . . 3 𝐹 = (𝑥𝐴𝐵)
5 funcnvmpt.1 . . . 4 𝑥𝐴
6 nfcv 2907 . . . 4 𝑖𝐵
7 nfcsb1v 3857 . . . 4 𝑥𝑖 / 𝑥𝐵
8 csbeq1a 3846 . . . 4 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
95, 2, 6, 7, 8cbvmptf 5183 . . 3 (𝑥𝐴𝐵) = (𝑖𝐴𝑖 / 𝑥𝐵)
104, 9eqtri 2766 . 2 𝐹 = (𝑖𝐴𝑖 / 𝑥𝐵)
11 funcnvmpt.4 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
1211sbimi 2077 . . 3 ([𝑖 / 𝑥](𝜑𝑥𝐴) → [𝑖 / 𝑥]𝐵𝑉)
13 funcnvmpt.0 . . . . 5 𝑥𝜑
14 nfcv 2907 . . . . . 6 𝑥𝑖
1514, 5nfel 2921 . . . . 5 𝑥 𝑖𝐴
1613, 15nfan 1902 . . . 4 𝑥(𝜑𝑖𝐴)
17 eleq1w 2821 . . . . 5 (𝑥 = 𝑖 → (𝑥𝐴𝑖𝐴))
1817anbi2d 629 . . . 4 (𝑥 = 𝑖 → ((𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴)))
1916, 18sbiev 2309 . . 3 ([𝑖 / 𝑥](𝜑𝑥𝐴) ↔ (𝜑𝑖𝐴))
20 nfcv 2907 . . . . 5 𝑥𝑉
217, 20nfel 2921 . . . 4 𝑥𝑖 / 𝑥𝐵𝑉
228eleq1d 2823 . . . 4 (𝑥 = 𝑖 → (𝐵𝑉𝑖 / 𝑥𝐵𝑉))
2321, 22sbiev 2309 . . 3 ([𝑖 / 𝑥]𝐵𝑉𝑖 / 𝑥𝐵𝑉)
2412, 19, 233imtr3i 291 . 2 ((𝜑𝑖𝐴) → 𝑖 / 𝑥𝐵𝑉)
25 csbeq1 3835 . 2 (𝑖 = 𝑗𝑖 / 𝑥𝐵 = 𝑗 / 𝑥𝐵)
261, 2, 3, 10, 24, 25funcnv5mpt 31005 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844   = wceq 1539  wnf 1786  [wsb 2067  wcel 2106  wnfc 2887  wne 2943  wral 3064  csb 3832  cmpt 5157  ccnv 5588  Fun wfun 6427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441
This theorem is referenced by:  disjdsct  31035
  Copyright terms: Public domain W3C validator