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Theorem mptfnf 6224
Description: The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Thierry Arnoux, 10-May-2017.)
Hypothesis
Ref Expression
mptfnf.0 𝑥𝐴
Assertion
Ref Expression
mptfnf (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)

Proof of Theorem mptfnf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eueq 3573 . . 3 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
21ralbii 3166 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
3 r19.26 3250 . . 3 (∀𝑥𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
4 eu5 2656 . . . 4 (∃!𝑦 𝑦 = 𝐵 ↔ (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
54ralbii 3166 . . 3 (∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ ∀𝑥𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
6 df-mpt 4922 . . . . . 6 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
76fneq1i 6194 . . . . 5 ((𝑥𝐴𝐵) Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} Fn 𝐴)
8 df-fn 6102 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
97, 8bitri 266 . . . 4 ((𝑥𝐴𝐵) Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
10 moanimv 2693 . . . . . . 7 (∃*𝑦(𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
1110albii 1904 . . . . . 6 (∀𝑥∃*𝑦(𝑥𝐴𝑦 = 𝐵) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
12 funopab 6134 . . . . . 6 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦 = 𝐵))
13 df-ral 3099 . . . . . 6 (∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
1411, 12, 133bitr4ri 295 . . . . 5 (∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)})
15 eqcom 2811 . . . . . 6 ({𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
16 dmopab 5534 . . . . . . . 8 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 = 𝐵)}
17 19.42v 2044 . . . . . . . . 9 (∃𝑦(𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
1817abbii 2921 . . . . . . . 8 {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
1916, 18eqtri 2826 . . . . . . 7 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
2019eqeq1i 2809 . . . . . 6 (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴)
21 pm4.71 549 . . . . . . . 8 ((𝑥𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
2221albii 1904 . . . . . . 7 (∀𝑥(𝑥𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
23 df-ral 3099 . . . . . . 7 (∀𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥𝐴 → ∃𝑦 𝑦 = 𝐵))
24 mptfnf.0 . . . . . . . 8 𝑥𝐴
2524abeq2f 2974 . . . . . . 7 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
2622, 23, 253bitr4i 294 . . . . . 6 (∀𝑥𝐴𝑦 𝑦 = 𝐵𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
2715, 20, 263bitr4ri 295 . . . . 5 (∀𝑥𝐴𝑦 𝑦 = 𝐵 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴)
2814, 27anbi12i 614 . . . 4 ((∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴𝑦 𝑦 = 𝐵) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
29 ancom 450 . . . 4 ((∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
309, 28, 293bitr2i 290 . . 3 ((𝑥𝐴𝐵) Fn 𝐴 ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
313, 5, 303bitr4ri 295 . 2 ((𝑥𝐴𝐵) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
322, 31bitr4i 269 1 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2156  ∃!weu 2630  ∃*wmo 2631  {cab 2790  wnfc 2933  wral 3094  Vcvv 3389  {copab 4904  cmpt 4921  dom cdm 5309  Fun wfun 6093   Fn wfn 6094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-sep 4973  ax-nul 4981  ax-pr 5094
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ral 3099  df-rab 3103  df-v 3391  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4115  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5217  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-fun 6101  df-fn 6102
This theorem is referenced by:  fnmptf  6225  mptfnd  39933
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