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Theorem dffo4 5258
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 5053 . . 3 (𝐹:AontoB ↔ (𝐹:AB ran 𝐹 = B))
2 simpl 102 . . . 4 ((𝐹:AB ran 𝐹 = B) → 𝐹:AB)
3 vex 2554 . . . . . . . . . 10 y V
43elrn 4520 . . . . . . . . 9 (y ran 𝐹x x𝐹y)
5 eleq2 2098 . . . . . . . . 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 445 . . . . . 6 (((𝐹:AB ran 𝐹 = B) y B) → x x𝐹y)
9 ffn 4989 . . . . . . . . . . 11 (𝐹:AB𝐹 Fn A)
10 fnbr 4944 . . . . . . . . . . . 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 1757 . . . . . . . 8 (𝐹:AB → (x x𝐹yx(x A x𝐹y)))
15 df-rex 2306 . . . . . . . 8 (x A x𝐹yx(x A x𝐹y))
1614, 15syl6ibr 151 . . . . . . 7 (𝐹:AB → (x x𝐹yx A x𝐹y))
1716ad2antrr 457 . . . . . 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 2386 . . . 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 5157 . . . . . . . . 9 ((𝐹 Fn A x A) → ((𝐹x) = yx𝐹y))
2322biimprd 147 . . . . . . . 8 ((𝐹 Fn A x A) → (x𝐹y → (𝐹x) = y))
24 eqcom 2039 . . . . . . . 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 2415 . . . . 5 (𝐹:AB → (x A x𝐹yx A y = (𝐹x)))
2827ralimdv 2382 . . . 4 (𝐹:AB → (y B x A x𝐹yy B x A y = (𝐹x)))
2928imdistani 419 . . 3 ((𝐹:AB y B x A x𝐹y) → (𝐹:AB y B x A y = (𝐹x)))
30 dffo3 5257 . . 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 1242  wex 1378   wcel 1390  wral 2300  wrex 2301   class class class wbr 3755  ran crn 4289   Fn wfn 4840  wf 4841  ontowfo 4843  cfv 4845
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-bndl 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-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853
This theorem is referenced by:  dffo5  5259
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