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

Theorem funcnvmpt 29852
Description: Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)
Hypotheses
Ref Expression
funcnvmpt.0 𝑥𝜑
funcnvmpt.1 𝑥𝐴
funcnvmpt.2 𝑥𝐹
funcnvmpt.3 𝐹 = (𝑥𝐴𝐵)
funcnvmpt.4 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
funcnvmpt (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐹   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem funcnvmpt
StepHypRef Expression
1 relcnv 5685 . . . 4 Rel 𝐹
2 nfcv 2907 . . . . 5 𝑦𝐹
3 funcnvmpt.2 . . . . . 6 𝑥𝐹
43nfcnv 5469 . . . . 5 𝑥𝐹
52, 4dffun6f 6082 . . . 4 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑦∃*𝑥 𝑦𝐹𝑥))
61, 5mpbiran 700 . . 3 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑦𝐹𝑥)
7 vex 3353 . . . . . 6 𝑦 ∈ V
8 vex 3353 . . . . . 6 𝑥 ∈ V
97, 8brcnv 5473 . . . . 5 (𝑦𝐹𝑥𝑥𝐹𝑦)
109mobii 2568 . . . 4 (∃*𝑥 𝑦𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
1110albii 1914 . . 3 (∀𝑦∃*𝑥 𝑦𝐹𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
126, 11bitri 266 . 2 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
13 funcnvmpt.0 . . . . 5 𝑥𝜑
14 funcnvmpt.3 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
1514funmpt2 6107 . . . . . . . . 9 Fun 𝐹
16 funbrfv2b 6429 . . . . . . . . 9 (Fun 𝐹 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) = 𝑦)))
1715, 16ax-mp 5 . . . . . . . 8 (𝑥𝐹𝑦 ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) = 𝑦))
18 funcnvmpt.4 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → 𝐵𝑉)
1918elexd 3367 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐵 ∈ V)
2019ex 401 . . . . . . . . . . . . 13 (𝜑 → (𝑥𝐴𝐵 ∈ V))
2113, 20ralrimi 3104 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝐴 𝐵 ∈ V)
22 funcnvmpt.1 . . . . . . . . . . . . 13 𝑥𝐴
2322rabid2f 3267 . . . . . . . . . . . 12 (𝐴 = {𝑥𝐴𝐵 ∈ V} ↔ ∀𝑥𝐴 𝐵 ∈ V)
2421, 23sylibr 225 . . . . . . . . . . 11 (𝜑𝐴 = {𝑥𝐴𝐵 ∈ V})
2514dmmpt 5816 . . . . . . . . . . 11 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
2624, 25syl6reqr 2818 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
2726eleq2d 2830 . . . . . . . . 9 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2827anbi1d 623 . . . . . . . 8 (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) = 𝑦) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
2917, 28syl5bb 274 . . . . . . 7 (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
3029bian1d 29697 . . . . . 6 (𝜑 → ((𝑥𝐴𝑥𝐹𝑦) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
31 simpr 477 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
3214fveq1i 6376 . . . . . . . . . . 11 (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥)
3322fvmpt2f 6472 . . . . . . . . . . 11 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3432, 33syl5eq 2811 . . . . . . . . . 10 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
3531, 18, 34syl2anc 579 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
3635eqeq2d 2775 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑦 = 𝐵))
37 eqcom 2772 . . . . . . . . 9 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
3827biimpar 469 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐹)
39 funbrfvb 6426 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
4015, 38, 39sylancr 581 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
4137, 40syl5bbr 276 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
4236, 41bitr3d 272 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑦 = 𝐵𝑥𝐹𝑦))
4342pm5.32da 574 . . . . . 6 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑥𝐹𝑦)))
4430, 43, 293bitr4rd 303 . . . . 5 (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = 𝐵)))
4513, 44mobid 2580 . . . 4 (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥(𝑥𝐴𝑦 = 𝐵)))
46 df-rmo 3063 . . . 4 (∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∃*𝑥(𝑥𝐴𝑦 = 𝐵))
4745, 46syl6bbr 280 . . 3 (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥𝐴 𝑦 = 𝐵))
4847albidv 2015 . 2 (𝜑 → (∀𝑦∃*𝑥 𝑥𝐹𝑦 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
4912, 48syl5bb 274 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wnf 1878  wcel 2155  ∃*wmo 2563  wnfc 2894  wral 3055  ∃*wrmo 3058  {crab 3059  Vcvv 3350   class class class wbr 4809  cmpt 4888  ccnv 5276  dom cdm 5277  Rel wrel 5282  Fun wfun 6062  cfv 6068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076
This theorem is referenced by:  funcnv5mpt  29853
  Copyright terms: Public domain W3C validator