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Theorem funcnvmpt 32684
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 6125 . . . 4 Rel 𝐹
2 nfcv 2903 . . . . 5 𝑦𝐹
3 funcnvmpt.2 . . . . . 6 𝑥𝐹
43nfcnv 5892 . . . . 5 𝑥𝐹
52, 4dffun6f 6581 . . . 4 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑦∃*𝑥 𝑦𝐹𝑥))
61, 5mpbiran 709 . . 3 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑦𝐹𝑥)
7 vex 3482 . . . . . 6 𝑦 ∈ V
8 vex 3482 . . . . . 6 𝑥 ∈ V
97, 8brcnv 5896 . . . . 5 (𝑦𝐹𝑥𝑥𝐹𝑦)
109mobii 2546 . . . 4 (∃*𝑥 𝑦𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
1110albii 1816 . . 3 (∀𝑦∃*𝑥 𝑦𝐹𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
126, 11bitri 275 . 2 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦)
13 funcnvmpt.0 . . . . 5 𝑥𝜑
14 funcnvmpt.3 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
1514funmpt2 6607 . . . . . . . . 9 Fun 𝐹
16 funbrfv2b 6966 . . . . . . . . 9 (Fun 𝐹 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) = 𝑦)))
1715, 16ax-mp 5 . . . . . . . 8 (𝑥𝐹𝑦 ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) = 𝑦))
1814dmmpt 6262 . . . . . . . . . . 11 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
19 funcnvmpt.4 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → 𝐵𝑉)
2019elexd 3502 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐵 ∈ V)
2120ex 412 . . . . . . . . . . . . 13 (𝜑 → (𝑥𝐴𝐵 ∈ V))
2213, 21ralrimi 3255 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝐴 𝐵 ∈ V)
23 funcnvmpt.1 . . . . . . . . . . . . 13 𝑥𝐴
2423rabid2f 3466 . . . . . . . . . . . 12 (𝐴 = {𝑥𝐴𝐵 ∈ V} ↔ ∀𝑥𝐴 𝐵 ∈ V)
2522, 24sylibr 234 . . . . . . . . . . 11 (𝜑𝐴 = {𝑥𝐴𝐵 ∈ V})
2618, 25eqtr4id 2794 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
2726eleq2d 2825 . . . . . . . . 9 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2827anbi1d 631 . . . . . . . 8 (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) = 𝑦) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
2917, 28bitrid 283 . . . . . . 7 (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
3029bian1d 32484 . . . . . 6 (𝜑 → ((𝑥𝐴𝑥𝐹𝑦) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
31 simpr 484 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
3214fveq1i 6908 . . . . . . . . . . 11 (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥)
3323fvmpt2f 7017 . . . . . . . . . . 11 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3432, 33eqtrid 2787 . . . . . . . . . 10 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
3531, 19, 34syl2anc 584 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
3635eqeq2d 2746 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑦 = 𝐵))
37 eqcom 2742 . . . . . . . . 9 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
3827biimpar 477 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐹)
39 funbrfvb 6962 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
4015, 38, 39sylancr 587 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
4137, 40bitr3id 285 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
4236, 41bitr3d 281 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑦 = 𝐵𝑥𝐹𝑦))
4342pm5.32da 579 . . . . . 6 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑥𝐹𝑦)))
4430, 43, 293bitr4rd 312 . . . . 5 (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = 𝐵)))
4513, 44mobid 2548 . . . 4 (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥(𝑥𝐴𝑦 = 𝐵)))
46 df-rmo 3378 . . . 4 (∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∃*𝑥(𝑥𝐴𝑦 = 𝐵))
4745, 46bitr4di 289 . . 3 (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥𝐴 𝑦 = 𝐵))
4847albidv 1918 . 2 (𝜑 → (∀𝑦∃*𝑥 𝑥𝐹𝑦 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
4912, 48bitrid 283 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wnf 1780  wcel 2106  ∃*wmo 2536  wnfc 2888  wral 3059  ∃*wrmo 3377  {crab 3433  Vcvv 3478   class class class wbr 5148  cmpt 5231  ccnv 5688  dom cdm 5689  Rel wrel 5694  Fun wfun 6557  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  funcnv5mpt  32685
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