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Theorem dffun6f 4858
 Description: Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dffun6f.1 xA
dffun6f.2 yA
Assertion
Ref Expression
dffun6f (Fun A ↔ (Rel A x∃*y xAy))
Distinct variable group:   x,y
Allowed substitution hints:   A(x,y)

Proof of Theorem dffun6f
Dummy variables w v u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun2 4855 . 2 (Fun A ↔ (Rel A wvu((wAv wAu) → v = u)))
2 nfcv 2175 . . . . . . 7 yw
3 dffun6f.2 . . . . . . 7 yA
4 nfcv 2175 . . . . . . 7 yv
52, 3, 4nfbr 3799 . . . . . 6 y wAv
6 nfv 1418 . . . . . 6 v wAy
7 breq2 3759 . . . . . 6 (v = y → (wAvwAy))
85, 6, 7cbvmo 1937 . . . . 5 (∃*v wAv∃*y wAy)
98albii 1356 . . . 4 (w∃*v wAvw∃*y wAy)
10 breq2 3759 . . . . . 6 (v = u → (wAvwAu))
1110mo4 1958 . . . . 5 (∃*v wAvvu((wAv wAu) → v = u))
1211albii 1356 . . . 4 (w∃*v wAvwvu((wAv wAu) → v = u))
13 nfcv 2175 . . . . . . 7 xw
14 dffun6f.1 . . . . . . 7 xA
15 nfcv 2175 . . . . . . 7 xy
1613, 14, 15nfbr 3799 . . . . . 6 x wAy
1716nfmo 1917 . . . . 5 x∃*y wAy
18 nfv 1418 . . . . 5 w∃*y xAy
19 breq1 3758 . . . . . 6 (w = x → (wAyxAy))
2019mobidv 1933 . . . . 5 (w = x → (∃*y wAy∃*y xAy))
2117, 18, 20cbval 1634 . . . 4 (w∃*y wAyx∃*y xAy)
229, 12, 213bitr3ri 200 . . 3 (x∃*y xAywvu((wAv wAu) → v = u))
2322anbi2i 430 . 2 ((Rel A x∃*y xAy) ↔ (Rel A wvu((wAv wAu) → v = u)))
241, 23bitr4i 176 1 (Fun A ↔ (Rel A x∃*y xAy))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃*wmo 1898  Ⅎwnfc 2162   class class class wbr 3755  Rel wrel 4293  Fun wfun 4839 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-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-cnv 4296  df-co 4297  df-fun 4847 This theorem is referenced by:  dffun6  4859  dffun4f  4861  funopab  4878
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