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Theorem mptfnf 6672
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 3700 . . 3 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
21ralbii 3092 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
3 r19.26 3110 . . 3 (∀𝑥𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
4 df-eu 2562 . . . 4 (∃!𝑦 𝑦 = 𝐵 ↔ (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
54ralbii 3092 . . 3 (∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ ∀𝑥𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
6 df-mpt 5225 . . . . . 6 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
76fneq1i 6635 . . . . 5 ((𝑥𝐴𝐵) Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} Fn 𝐴)
8 df-fn 6535 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
97, 8bitri 274 . . . 4 ((𝑥𝐴𝐵) Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
10 moanimv 2614 . . . . . . 7 (∃*𝑦(𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
1110albii 1821 . . . . . 6 (∀𝑥∃*𝑦(𝑥𝐴𝑦 = 𝐵) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
12 funopab 6572 . . . . . 6 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦 = 𝐵))
13 df-ral 3061 . . . . . 6 (∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
1411, 12, 133bitr4ri 303 . . . . 5 (∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)})
15 eqcom 2738 . . . . . 6 ({𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
16 dmopab 5907 . . . . . . . 8 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 = 𝐵)}
17 19.42v 1957 . . . . . . . . 9 (∃𝑦(𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
1817abbii 2801 . . . . . . . 8 {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
1916, 18eqtri 2759 . . . . . . 7 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
2019eqeq1i 2736 . . . . . 6 (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴)
21 pm4.71 558 . . . . . . . 8 ((𝑥𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
2221albii 1821 . . . . . . 7 (∀𝑥(𝑥𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
23 df-ral 3061 . . . . . . 7 (∀𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥𝐴 → ∃𝑦 𝑦 = 𝐵))
24 mptfnf.0 . . . . . . . 8 𝑥𝐴
2524eqabf 2934 . . . . . . 7 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
2622, 23, 253bitr4i 302 . . . . . 6 (∀𝑥𝐴𝑦 𝑦 = 𝐵𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
2715, 20, 263bitr4ri 303 . . . . 5 (∀𝑥𝐴𝑦 𝑦 = 𝐵 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴)
2814, 27anbi12i 627 . . . 4 ((∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴𝑦 𝑦 = 𝐵) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
29 ancom 461 . . . 4 ((∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
309, 28, 293bitr2i 298 . . 3 ((𝑥𝐴𝐵) Fn 𝐴 ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
313, 5, 303bitr4ri 303 . 2 ((𝑥𝐴𝐵) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
322, 31bitr4i 277 1 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  ∃*wmo 2531  ∃!weu 2561  {cab 2708  wnfc 2882  wral 3060  Vcvv 3473  {copab 5203  cmpt 5224  dom cdm 5669  Fun wfun 6526   Fn wfn 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-fun 6534  df-fn 6535
This theorem is referenced by:  fnmptf  6673  mptfnd  43717  fnmptif  43743
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