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Theorem mptfnf 5911
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 3341 . . 3 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
21ralbii 2959 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
3 r19.26 3042 . . 3 (∀𝑥𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵) ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
4 eu5 2480 . . . 4 (∃!𝑦 𝑦 = 𝐵 ↔ (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
54ralbii 2959 . . 3 (∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ ∀𝑥𝐴 (∃𝑦 𝑦 = 𝐵 ∧ ∃*𝑦 𝑦 = 𝐵))
6 df-mpt 4636 . . . . . 6 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
76fneq1i 5882 . . . . 5 ((𝑥𝐴𝐵) Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} Fn 𝐴)
8 df-fn 5790 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
97, 8bitri 262 . . . 4 ((𝑥𝐴𝐵) Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴))
10 moanimv 2515 . . . . . . 7 (∃*𝑦(𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
1110albii 1736 . . . . . 6 (∀𝑥∃*𝑦(𝑥𝐴𝑦 = 𝐵) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
12 funopab 5820 . . . . . 6 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦 = 𝐵))
13 df-ral 2897 . . . . . 6 (∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦 𝑦 = 𝐵))
1411, 12, 133bitr4ri 291 . . . . 5 (∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)})
15 eqcom 2613 . . . . . 6 ({𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
16 dmopab 5241 . . . . . . . 8 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 = 𝐵)}
17 19.42v 1904 . . . . . . . . 9 (∃𝑦(𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
1817abbii 2722 . . . . . . . 8 {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
1916, 18eqtri 2628 . . . . . . 7 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)}
2019eqeq1i 2611 . . . . . 6 (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} = 𝐴)
21 pm4.71 659 . . . . . . . 8 ((𝑥𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
2221albii 1736 . . . . . . 7 (∀𝑥(𝑥𝐴 → ∃𝑦 𝑦 = 𝐵) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
23 df-ral 2897 . . . . . . 7 (∀𝑥𝐴𝑦 𝑦 = 𝐵 ↔ ∀𝑥(𝑥𝐴 → ∃𝑦 𝑦 = 𝐵))
24 mptfnf.0 . . . . . . . 8 𝑥𝐴
2524abeq2f 2774 . . . . . . 7 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)))
2622, 23, 253bitr4i 290 . . . . . 6 (∀𝑥𝐴𝑦 𝑦 = 𝐵𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 = 𝐵)})
2715, 20, 263bitr4ri 291 . . . . 5 (∀𝑥𝐴𝑦 𝑦 = 𝐵 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = 𝐴)
2814, 27anbi12i 728 . . . 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 286 . . 3 ((𝑥𝐴𝐵) Fn 𝐴 ↔ (∀𝑥𝐴𝑦 𝑦 = 𝐵 ∧ ∀𝑥𝐴 ∃*𝑦 𝑦 = 𝐵))
313, 5, 303bitr4ri 291 . 2 ((𝑥𝐴𝐵) Fn 𝐴 ↔ ∀𝑥𝐴 ∃!𝑦 𝑦 = 𝐵)
322, 31bitr4i 265 1 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1976  ∃!weu 2454  ∃*wmo 2455  {cab 2592  wnfc 2734  wral 2892  Vcvv 3169  {copab 4633  cmpt 4634  dom cdm 5025  Fun wfun 5781   Fn wfn 5782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-fun 5789  df-fn 5790
This theorem is referenced by:  fnmptf  5912
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