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 30404
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 5960 . . . 4 Rel 𝐹
2 nfcv 2975 . . . . 5 𝑦𝐹
3 funcnvmpt.2 . . . . . 6 𝑥𝐹
43nfcnv 5742 . . . . 5 𝑥𝐹
52, 4dffun6f 6362 . . . 4 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑦∃*𝑥 𝑦𝐹𝑥))
61, 5mpbiran 707 . . 3 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑦𝐹𝑥)
7 vex 3496 . . . . . 6 𝑦 ∈ V
8 vex 3496 . . . . . 6 𝑥 ∈ V
97, 8brcnv 5746 . . . . 5 (𝑦𝐹𝑥𝑥𝐹𝑦)
109mobii 2625 . . . 4 (∃*𝑥 𝑦𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
1110albii 1814 . . 3 (∀𝑦∃*𝑥 𝑦𝐹𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
126, 11bitri 277 . 2 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
13 funcnvmpt.0 . . . . 5 𝑥𝜑
14 funcnvmpt.3 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
1514funmpt2 6387 . . . . . . . . 9 Fun 𝐹
16 funbrfv2b 6716 . . . . . . . . 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 3513 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐵 ∈ V)
2019ex 415 . . . . . . . . . . . . 13 (𝜑 → (𝑥𝐴𝐵 ∈ V))
2113, 20ralrimi 3214 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝐴 𝐵 ∈ V)
22 funcnvmpt.1 . . . . . . . . . . . . 13 𝑥𝐴
2322rabid2f 3381 . . . . . . . . . . . 12 (𝐴 = {𝑥𝐴𝐵 ∈ V} ↔ ∀𝑥𝐴 𝐵 ∈ V)
2421, 23sylibr 236 . . . . . . . . . . 11 (𝜑𝐴 = {𝑥𝐴𝐵 ∈ V})
2514dmmpt 6087 . . . . . . . . . . 11 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
2624, 25syl6reqr 2873 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
2726eleq2d 2896 . . . . . . . . 9 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2827anbi1d 631 . . . . . . . 8 (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) = 𝑦) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
2917, 28syl5bb 285 . . . . . . 7 (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
3029bian1d 30216 . . . . . 6 (𝜑 → ((𝑥𝐴𝑥𝐹𝑦) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
31 simpr 487 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
3214fveq1i 6664 . . . . . . . . . . 11 (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥)
3322fvmpt2f 6762 . . . . . . . . . . 11 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3432, 33syl5eq 2866 . . . . . . . . . 10 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
3531, 18, 34syl2anc 586 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
3635eqeq2d 2830 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑦 = 𝐵))
37 eqcom 2826 . . . . . . . . 9 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
3827biimpar 480 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐹)
39 funbrfvb 6713 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
4015, 38, 39sylancr 589 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
4137, 40syl5bbr 287 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
4236, 41bitr3d 283 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑦 = 𝐵𝑥𝐹𝑦))
4342pm5.32da 581 . . . . . 6 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑥𝐹𝑦)))
4430, 43, 293bitr4rd 314 . . . . 5 (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = 𝐵)))
4513, 44mobid 2628 . . . 4 (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥(𝑥𝐴𝑦 = 𝐵)))
46 df-rmo 3144 . . . 4 (∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∃*𝑥(𝑥𝐴𝑦 = 𝐵))
4745, 46syl6bbr 291 . . 3 (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥𝐴 𝑦 = 𝐵))
4847albidv 1915 . 2 (𝜑 → (∀𝑦∃*𝑥 𝑥𝐹𝑦 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
4912, 48syl5bb 285 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1529   = wceq 1531  wnf 1778  wcel 2108  ∃*wmo 2614  wnfc 2959  wral 3136  ∃*wrmo 3139  {crab 3140  Vcvv 3493   class class class wbr 5057  cmpt 5137  ccnv 5547  dom cdm 5548  Rel wrel 5553  Fun wfun 6342  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by:  funcnv5mpt  30405
  Copyright terms: Public domain W3C validator