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Theorem dffo4 5240
Description: Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
Assertion
Ref Expression
dffo4 (𝐹:AontoB ↔ (𝐹:AB y B x A x𝐹y))
Distinct variable groups:   x,y,A   x,B,y   x,𝐹,y

Proof of Theorem dffo4
StepHypRef Expression
1 dffo2 5035 . . 3 (𝐹:AontoB ↔ (𝐹:AB ran 𝐹 = B))
2 simpl 102 . . . 4 ((𝐹:AB ran 𝐹 = B) → 𝐹:AB)
3 vex 2538 . . . . . . . . . 10 y V
43elrn 4504 . . . . . . . . 9 (y ran 𝐹x x𝐹y)
5 eleq2 2083 . . . . . . . . 9 (ran 𝐹 = B → (y ran 𝐹y B))
64, 5syl5bbr 183 . . . . . . . 8 (ran 𝐹 = B → (x x𝐹yy B))
76biimpar 281 . . . . . . 7 ((ran 𝐹 = B y B) → x x𝐹y)
87adantll 448 . . . . . 6 (((𝐹:AB ran 𝐹 = B) y B) → x x𝐹y)
9 ffn 4972 . . . . . . . . . . 11 (𝐹:AB𝐹 Fn A)
10 fnbr 4927 . . . . . . . . . . . 12 ((𝐹 Fn A x𝐹y) → x A)
1110ex 108 . . . . . . . . . . 11 (𝐹 Fn A → (x𝐹yx A))
129, 11syl 14 . . . . . . . . . 10 (𝐹:AB → (x𝐹yx A))
1312ancrd 309 . . . . . . . . 9 (𝐹:AB → (x𝐹y → (x A x𝐹y)))
1413eximdv 1742 . . . . . . . 8 (𝐹:AB → (x x𝐹yx(x A x𝐹y)))
15 df-rex 2290 . . . . . . . 8 (x A x𝐹yx(x A x𝐹y))
1614, 15syl6ibr 151 . . . . . . 7 (𝐹:AB → (x x𝐹yx A x𝐹y))
1716ad2antrr 460 . . . . . 6 (((𝐹:AB ran 𝐹 = B) y B) → (x x𝐹yx A x𝐹y))
188, 17mpd 13 . . . . 5 (((𝐹:AB ran 𝐹 = B) y B) → x A x𝐹y)
1918ralrimiva 2370 . . . 4 ((𝐹:AB ran 𝐹 = B) → y B x A x𝐹y)
202, 19jca 290 . . 3 ((𝐹:AB ran 𝐹 = B) → (𝐹:AB y B x A x𝐹y))
211, 20sylbi 114 . 2 (𝐹:AontoB → (𝐹:AB y B x A x𝐹y))
22 fnbrfvb 5139 . . . . . . . . 9 ((𝐹 Fn A x A) → ((𝐹x) = yx𝐹y))
2322biimprd 147 . . . . . . . 8 ((𝐹 Fn A x A) → (x𝐹y → (𝐹x) = y))
24 eqcom 2024 . . . . . . . 8 ((𝐹x) = yy = (𝐹x))
2523, 24syl6ib 150 . . . . . . 7 ((𝐹 Fn A x A) → (x𝐹yy = (𝐹x)))
269, 25sylan 267 . . . . . 6 ((𝐹:AB x A) → (x𝐹yy = (𝐹x)))
2726reximdva 2399 . . . . 5 (𝐹:AB → (x A x𝐹yx A y = (𝐹x)))
2827ralimdv 2366 . . . 4 (𝐹:AB → (y B x A x𝐹yy B x A y = (𝐹x)))
2928imdistani 422 . . 3 ((𝐹:AB y B x A x𝐹y) → (𝐹:AB y B x A y = (𝐹x)))
30 dffo3 5239 . . 3 (𝐹:AontoB ↔ (𝐹:AB y B x A y = (𝐹x)))
3129, 30sylibr 137 . 2 ((𝐹:AB y B x A x𝐹y) → 𝐹:AontoB)
3221, 31impbii 117 1 (𝐹:AontoB ↔ (𝐹:AB y B x A x𝐹y))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  wral 2284  wrex 2285   class class class wbr 3738  ran crn 4273   Fn wfn 4824  wf 4825  ontowfo 4827  cfv 4829
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837
This theorem is referenced by:  dffo5  5241
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