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Theorem fnopabg 4965
 Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Hypothesis
Ref Expression
fnopabg.1 𝐹 = {⟨x, y⟩ ∣ (x A φ)}
Assertion
Ref Expression
fnopabg (x A ∃!yφ𝐹 Fn A)
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)   𝐹(x,y)

Proof of Theorem fnopabg
StepHypRef Expression
1 moanimv 1972 . . . . . 6 (∃*y(x A φ) ↔ (x A∃*yφ))
21albii 1356 . . . . 5 (x∃*y(x A φ) ↔ x(x A∃*yφ))
3 funopab 4878 . . . . 5 (Fun {⟨x, y⟩ ∣ (x A φ)} ↔ x∃*y(x A φ))
4 df-ral 2305 . . . . 5 (x A ∃*yφx(x A∃*yφ))
52, 3, 43bitr4ri 202 . . . 4 (x A ∃*yφ ↔ Fun {⟨x, y⟩ ∣ (x A φ)})
6 dmopab3 4491 . . . 4 (x A yφ ↔ dom {⟨x, y⟩ ∣ (x A φ)} = A)
75, 6anbi12i 433 . . 3 ((x A ∃*yφ x A yφ) ↔ (Fun {⟨x, y⟩ ∣ (x A φ)} dom {⟨x, y⟩ ∣ (x A φ)} = A))
8 r19.26 2435 . . 3 (x A (∃*yφ yφ) ↔ (x A ∃*yφ x A yφ))
9 df-fn 4848 . . 3 ({⟨x, y⟩ ∣ (x A φ)} Fn A ↔ (Fun {⟨x, y⟩ ∣ (x A φ)} dom {⟨x, y⟩ ∣ (x A φ)} = A))
107, 8, 93bitr4i 201 . 2 (x A (∃*yφ yφ) ↔ {⟨x, y⟩ ∣ (x A φ)} Fn A)
11 eu5 1944 . . . 4 (∃!yφ ↔ (yφ ∃*yφ))
12 ancom 253 . . . 4 ((yφ ∃*yφ) ↔ (∃*yφ yφ))
1311, 12bitri 173 . . 3 (∃!yφ ↔ (∃*yφ yφ))
1413ralbii 2324 . 2 (x A ∃!yφx A (∃*yφ yφ))
15 fnopabg.1 . . 3 𝐹 = {⟨x, y⟩ ∣ (x A φ)}
1615fneq1i 4936 . 2 (𝐹 Fn A ↔ {⟨x, y⟩ ∣ (x A φ)} Fn A)
1710, 14, 163bitr4i 201 1 (x A ∃!yφ𝐹 Fn A)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  ∃*wmo 1898  ∀wral 2300  {copab 3808  dom cdm 4288  Fun wfun 4839   Fn wfn 4840 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-fn 4848 This theorem is referenced by:  fnopab  4966  mptfng  4967
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