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Theorem fnopabg 4944
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 1953 . . . . . 6 (∃*y(x A φ) ↔ (x A∃*yφ))
21albii 1335 . . . . 5 (x∃*y(x A φ) ↔ x(x A∃*yφ))
3 funopab 4857 . . . . 5 (Fun {⟨x, y⟩ ∣ (x A φ)} ↔ x∃*y(x A φ))
4 df-ral 2285 . . . . 5 (x A ∃*yφx(x A∃*yφ))
52, 3, 43bitr4ri 202 . . . 4 (x A ∃*yφ ↔ Fun {⟨x, y⟩ ∣ (x A φ)})
6 dmopab3 4471 . . . 4 (x A yφ ↔ dom {⟨x, y⟩ ∣ (x A φ)} = A)
75, 6anbi12i 436 . . 3 ((x A ∃*yφ x A yφ) ↔ (Fun {⟨x, y⟩ ∣ (x A φ)} dom {⟨x, y⟩ ∣ (x A φ)} = A))
8 r19.26 2415 . . 3 (x A (∃*yφ yφ) ↔ (x A ∃*yφ x A yφ))
9 df-fn 4828 . . 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 1925 . . . 4 (∃!yφ ↔ (yφ ∃*yφ))
12 ancom 253 . . . 4 ((yφ ∃*yφ) ↔ (∃*yφ yφ))
1311, 12bitri 173 . . 3 (∃!yφ ↔ (∃*yφ yφ))
1413ralbii 2304 . 2 (x A ∃!yφx A (∃*yφ yφ))
15 fnopabg.1 . . 3 𝐹 = {⟨x, y⟩ ∣ (x A φ)}
1615fneq1i 4915 . 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 1224   = wceq 1226  wex 1358   wcel 1370  ∃!weu 1878  ∃*wmo 1879  wral 2280  {copab 3787  dom cdm 4268  Fun wfun 4819   Fn wfn 4820
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-fun 4827  df-fn 4828
This theorem is referenced by:  fnopab  4945  mptfng  4946
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