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Theorem funcnvmpt 29596
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 5538 . . . 4 Rel 𝐹
2 nfcv 2793 . . . . 5 𝑦𝐹
3 funcnvmpt.2 . . . . . 6 𝑥𝐹
43nfcnv 5333 . . . . 5 𝑥𝐹
52, 4dffun6f 5940 . . . 4 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑦∃*𝑥 𝑦𝐹𝑥))
61, 5mpbiran 973 . . 3 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑦𝐹𝑥)
7 vex 3234 . . . . . 6 𝑦 ∈ V
8 vex 3234 . . . . . 6 𝑥 ∈ V
97, 8brcnv 5337 . . . . 5 (𝑦𝐹𝑥𝑥𝐹𝑦)
109mobii 2521 . . . 4 (∃*𝑥 𝑦𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
1110albii 1787 . . 3 (∀𝑦∃*𝑥 𝑦𝐹𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
126, 11bitri 264 . 2 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
13 funcnvmpt.0 . . . . 5 𝑥𝜑
14 funcnvmpt.3 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
1514funmpt2 5965 . . . . . . . . 9 Fun 𝐹
16 funbrfv2b 6279 . . . . . . . . 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)
19 elex 3243 . . . . . . . . . . . . . . 15 (𝐵𝑉𝐵 ∈ V)
2018, 19syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐵 ∈ V)
2120ex 449 . . . . . . . . . . . . 13 (𝜑 → (𝑥𝐴𝐵 ∈ V))
2213, 21ralrimi 2986 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝐴 𝐵 ∈ V)
23 funcnvmpt.1 . . . . . . . . . . . . 13 𝑥𝐴
2423rabid2f 3149 . . . . . . . . . . . 12 (𝐴 = {𝑥𝐴𝐵 ∈ V} ↔ ∀𝑥𝐴 𝐵 ∈ V)
2522, 24sylibr 224 . . . . . . . . . . 11 (𝜑𝐴 = {𝑥𝐴𝐵 ∈ V})
2614dmmpt 5668 . . . . . . . . . . 11 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
2725, 26syl6reqr 2704 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
2827eleq2d 2716 . . . . . . . . 9 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2928anbi1d 741 . . . . . . . 8 (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) = 𝑦) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
3017, 29syl5bb 272 . . . . . . 7 (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
3130bian1d 29434 . . . . . 6 (𝜑 → ((𝑥𝐴𝑥𝐹𝑦) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
32 simpr 476 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
3314fveq1i 6230 . . . . . . . . . . 11 (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥)
3423fvmpt2f 6322 . . . . . . . . . . 11 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3533, 34syl5eq 2697 . . . . . . . . . 10 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
3632, 18, 35syl2anc 694 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
3736eqeq2d 2661 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑦 = 𝐵))
38 eqcom 2658 . . . . . . . . 9 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
3928biimpar 501 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐹)
40 funbrfvb 6276 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
4115, 39, 40sylancr 696 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
4238, 41syl5bbr 274 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
4337, 42bitr3d 270 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑦 = 𝐵𝑥𝐹𝑦))
4443pm5.32da 674 . . . . . 6 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑥𝐹𝑦)))
4531, 44, 303bitr4rd 301 . . . . 5 (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = 𝐵)))
4613, 45mobid 2517 . . . 4 (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥(𝑥𝐴𝑦 = 𝐵)))
47 df-rmo 2949 . . . 4 (∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∃*𝑥(𝑥𝐴𝑦 = 𝐵))
4846, 47syl6bbr 278 . . 3 (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥𝐴 𝑦 = 𝐵))
4948albidv 1889 . 2 (𝜑 → (∀𝑦∃*𝑥 𝑥𝐹𝑦 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
5012, 49syl5bb 272 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wnf 1748  wcel 2030  ∃*wmo 2499  wnfc 2780  wral 2941  ∃*wrmo 2944  {crab 2945  Vcvv 3231   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  Rel wrel 5148  Fun wfun 5920  cfv 5926
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-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-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-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:  funcnv5mpt  29597
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