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Theorem dffo4 5343
Description: Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
Assertion
Ref Expression
dffo4 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dffo4
StepHypRef Expression
1 dffo2 5138 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
2 simpl 106 . . . 4 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
3 vex 2577 . . . . . . . . . 10 𝑦 ∈ V
43elrn 4605 . . . . . . . . 9 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
5 eleq2 2117 . . . . . . . . 9 (ran 𝐹 = 𝐵 → (𝑦 ∈ ran 𝐹𝑦𝐵))
64, 5syl5bbr 187 . . . . . . . 8 (ran 𝐹 = 𝐵 → (∃𝑥 𝑥𝐹𝑦𝑦𝐵))
76biimpar 285 . . . . . . 7 ((ran 𝐹 = 𝐵𝑦𝐵) → ∃𝑥 𝑥𝐹𝑦)
87adantll 453 . . . . . 6 (((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦𝐵) → ∃𝑥 𝑥𝐹𝑦)
9 ffn 5074 . . . . . . . . . . 11 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
10 fnbr 5029 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
1110ex 112 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
129, 11syl 14 . . . . . . . . . 10 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦𝑥𝐴))
1312ancrd 313 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦 → (𝑥𝐴𝑥𝐹𝑦)))
1413eximdv 1776 . . . . . . . 8 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥𝐴𝑥𝐹𝑦)))
15 df-rex 2329 . . . . . . . 8 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
1614, 15syl6ibr 155 . . . . . . 7 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑥𝐹𝑦))
1716ad2antrr 465 . . . . . 6 (((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦𝐵) → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑥𝐹𝑦))
188, 17mpd 13 . . . . 5 (((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑥𝐹𝑦)
1918ralrimiva 2409 . . . 4 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) → ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦)
202, 19jca 294 . . 3 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
211, 20sylbi 118 . 2 (𝐹:𝐴onto𝐵 → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
22 fnbrfvb 5242 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
2322biimprd 151 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦))
24 eqcom 2058 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
2523, 24syl6ib 154 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑥𝐹𝑦𝑦 = (𝐹𝑥)))
269, 25sylan 271 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → (𝑥𝐹𝑦𝑦 = (𝐹𝑥)))
2726reximdva 2438 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥𝐴 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
2827ralimdv 2405 . . . 4 (𝐹:𝐴𝐵 → (∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
2928imdistani 427 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
30 dffo3 5342 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
3129, 30sylibr 141 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) → 𝐹:𝐴onto𝐵)
3221, 31impbii 121 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  wral 2323  wrex 2324   class class class wbr 3792  ran crn 4374   Fn wfn 4925  wf 4926  ontowfo 4928  cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938
This theorem is referenced by:  dffo5  5344
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