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Theorem funcnvmpt 29453
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 5501 . . . 4 Rel 𝐹
2 nfcv 2763 . . . . 5 𝑦𝐹
3 funcnvmpt.2 . . . . . 6 𝑥𝐹
43nfcnv 5299 . . . . 5 𝑥𝐹
52, 4dffun6f 5900 . . . 4 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑦∃*𝑥 𝑦𝐹𝑥))
61, 5mpbiran 953 . . 3 (Fun 𝐹 ↔ ∀𝑦∃*𝑥 𝑦𝐹𝑥)
7 vex 3201 . . . . . 6 𝑦 ∈ V
8 vex 3201 . . . . . 6 𝑥 ∈ V
97, 8brcnv 5303 . . . . 5 (𝑦𝐹𝑥𝑥𝐹𝑦)
109mobii 2492 . . . 4 (∃*𝑥 𝑦𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
1110albii 1746 . . 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 5925 . . . . . . . . 9 Fun 𝐹
16 funbrfv2b 6238 . . . . . . . . 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 3210 . . . . . . . . . . . . . . 15 (𝐵𝑉𝐵 ∈ V)
2018, 19syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐵 ∈ V)
2120ex 450 . . . . . . . . . . . . 13 (𝜑 → (𝑥𝐴𝐵 ∈ V))
2213, 21ralrimi 2956 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝐴 𝐵 ∈ V)
23 funcnvmpt.1 . . . . . . . . . . . . 13 𝑥𝐴
2423rabid2f 3117 . . . . . . . . . . . 12 (𝐴 = {𝑥𝐴𝐵 ∈ V} ↔ ∀𝑥𝐴 𝐵 ∈ V)
2522, 24sylibr 224 . . . . . . . . . . 11 (𝜑𝐴 = {𝑥𝐴𝐵 ∈ V})
2614dmmpt 5628 . . . . . . . . . . 11 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
2725, 26syl6reqr 2674 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
2827eleq2d 2686 . . . . . . . . 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 29290 . . . . . 6 (𝜑 → ((𝑥𝐴𝑥𝐹𝑦) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑦)))
32 simpr 477 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
3314fveq1i 6190 . . . . . . . . . . 11 (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥)
3423fvmpt2f 6281 . . . . . . . . . . 11 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
3533, 34syl5eq 2667 . . . . . . . . . 10 ((𝑥𝐴𝐵𝑉) → (𝐹𝑥) = 𝐵)
3632, 18, 35syl2anc 693 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
3736eqeq2d 2631 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑦 = 𝐵))
38 eqcom 2628 . . . . . . . . 9 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
3928biimpar 502 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐹)
40 funbrfvb 6236 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
4115, 39, 40sylancr 695 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
4238, 41syl5bbr 274 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
4337, 42bitr3d 270 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑦 = 𝐵𝑥𝐹𝑦))
4443pm5.32da 673 . . . . . 6 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑥𝐹𝑦)))
4531, 44, 303bitr4rd 301 . . . . 5 (𝜑 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = 𝐵)))
4613, 45mobid 2488 . . . 4 (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥(𝑥𝐴𝑦 = 𝐵)))
47 df-rmo 2919 . . . 4 (∃*𝑥𝐴 𝑦 = 𝐵 ↔ ∃*𝑥(𝑥𝐴𝑦 = 𝐵))
4846, 47syl6bbr 278 . . 3 (𝜑 → (∃*𝑥 𝑥𝐹𝑦 ↔ ∃*𝑥𝐴 𝑦 = 𝐵))
4948albidv 1848 . 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 384  wal 1480   = wceq 1482  wnf 1707  wcel 1989  ∃*wmo 2470  wnfc 2750  wral 2911  ∃*wrmo 2914  {crab 2915  Vcvv 3198   class class class wbr 4651  cmpt 4727  ccnv 5111  dom cdm 5112  Rel wrel 5117  Fun wfun 5880  cfv 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894
This theorem is referenced by:  funcnv5mpt  29454
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