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Theorem mptfnf 6703
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 3714 . . 3 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
21ralbii 3093 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
3 r19.26 3111 . . 3 (∀𝑥𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
4 df-eu 2569 . . . 4 (∃!𝑦 𝑦 = 𝐵 ↔ (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
54ralbii 3093 . . 3 (∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ ∀𝑥𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
6 df-mpt 5226 . . . . . 6 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
76fneq1i 6665 . . . . 5 ((𝑥𝐴𝐵) Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} Fn 𝐴)
8 df-fn 6564 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
97, 8bitri 275 . . . 4 ((𝑥𝐴𝐵) Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
10 moanimv 2619 . . . . . . 7 (∃*𝑦(𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
1110albii 1819 . . . . . 6 (∀𝑥∃*𝑦(𝑥𝐴𝑦 = 𝐵) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
12 funopab 6601 . . . . . 6 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦 = 𝐵))
13 df-ral 3062 . . . . . 6 (∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
1411, 12, 133bitr4ri 304 . . . . 5 (∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)})
15 eqcom 2744 . . . . . 6 ({𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
16 dmopab 5926 . . . . . . . 8 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 = 𝐵)}
17 19.42v 1953 . . . . . . . . 9 (∃𝑦(𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
1817abbii 2809 . . . . . . . 8 {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
1916, 18eqtri 2765 . . . . . . 7 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
2019eqeq1i 2742 . . . . . 6 (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴)
21 pm4.71 557 . . . . . . . 8 ((𝑥𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
2221albii 1819 . . . . . . 7 (∀𝑥(𝑥𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
23 df-ral 3062 . . . . . . 7 (∀𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥𝐴 → ∃𝑦 𝑦 = 𝐵))
24 mptfnf.0 . . . . . . . 8 𝑥𝐴
2524eqabf 2935 . . . . . . 7 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
2622, 23, 253bitr4i 303 . . . . . 6 (∀𝑥𝐴𝑦 𝑦 = 𝐵𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
2715, 20, 263bitr4ri 304 . . . . 5 (∀𝑥𝐴𝑦 𝑦 = 𝐵 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴)
2814, 27anbi12i 628 . . . 4 ((∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴𝑦 𝑦 = 𝐵) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
29 ancom 460 . . . 4 ((∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
309, 28, 293bitr2i 299 . . 3 ((𝑥𝐴𝐵) Fn 𝐴 ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
313, 5, 303bitr4ri 304 . 2 ((𝑥𝐴𝐵) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
322, 31bitr4i 278 1 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  ∃*wmo 2538  ∃!weu 2568  {cab 2714  wnfc 2890  wral 3061  Vcvv 3480  {copab 5205  cmpt 5225  dom cdm 5685  Fun wfun 6555   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-fun 6563  df-fn 6564
This theorem is referenced by:  fnmptf  6704  mptfnd  45248  fnmptif  45272
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