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Theorem mptfnf 6471
 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 3624 . . 3 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
21ralbii 3097 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
3 r19.26 3101 . . 3 (∀𝑥𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
4 df-eu 2588 . . . 4 (∃!𝑦 𝑦 = 𝐵 ↔ (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
54ralbii 3097 . . 3 (∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ ∀𝑥𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
6 df-mpt 5117 . . . . . 6 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
76fneq1i 6436 . . . . 5 ((𝑥𝐴𝐵) Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} Fn 𝐴)
8 df-fn 6343 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
97, 8bitri 278 . . . 4 ((𝑥𝐴𝐵) Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
10 moanimv 2640 . . . . . . 7 (∃*𝑦(𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
1110albii 1821 . . . . . 6 (∀𝑥∃*𝑦(𝑥𝐴𝑦 = 𝐵) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
12 funopab 6375 . . . . . 6 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦 = 𝐵))
13 df-ral 3075 . . . . . 6 (∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
1411, 12, 133bitr4ri 307 . . . . 5 (∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)})
15 eqcom 2765 . . . . . 6 ({𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
16 dmopab 5761 . . . . . . . 8 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 = 𝐵)}
17 19.42v 1954 . . . . . . . . 9 (∃𝑦(𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
1817abbii 2823 . . . . . . . 8 {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
1916, 18eqtri 2781 . . . . . . 7 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
2019eqeq1i 2763 . . . . . 6 (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴)
21 pm4.71 561 . . . . . . . 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 3075 . . . . . . 7 (∀𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥𝐴 → ∃𝑦 𝑦 = 𝐵))
24 mptfnf.0 . . . . . . . 8 𝑥𝐴
2524abeq2f 2949 . . . . . . 7 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
2622, 23, 253bitr4i 306 . . . . . 6 (∀𝑥𝐴𝑦 𝑦 = 𝐵𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
2715, 20, 263bitr4ri 307 . . . . 5 (∀𝑥𝐴𝑦 𝑦 = 𝐵 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴)
2814, 27anbi12i 629 . . . 4 ((∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴𝑦 𝑦 = 𝐵) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
29 ancom 464 . . . 4 ((∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
309, 28, 293bitr2i 302 . . 3 ((𝑥𝐴𝐵) Fn 𝐴 ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
313, 5, 303bitr4ri 307 . 2 ((𝑥𝐴𝐵) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
322, 31bitr4i 281 1 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃*wmo 2555  ∃!weu 2587  {cab 2735  Ⅎwnfc 2899  ∀wral 3070  Vcvv 3409  {copab 5098   ↦ cmpt 5116  dom cdm 5528  Fun wfun 6334   Fn wfn 6335 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-fun 6342  df-fn 6343 This theorem is referenced by:  fnmptf  6472  mptfnd  42281
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