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Theorem dffo4 5315
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 5110 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
2 simpl 102 . . . 4 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
3 vex 2560 . . . . . . . . . 10 𝑦 ∈ V
43elrn 4577 . . . . . . . . 9 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
5 eleq2 2101 . . . . . . . . 9 (ran 𝐹 = 𝐵 → (𝑦 ∈ ran 𝐹𝑦𝐵))
64, 5syl5bbr 183 . . . . . . . 8 (ran 𝐹 = 𝐵 → (∃𝑥 𝑥𝐹𝑦𝑦𝐵))
76biimpar 281 . . . . . . 7 ((ran 𝐹 = 𝐵𝑦𝐵) → ∃𝑥 𝑥𝐹𝑦)
87adantll 445 . . . . . 6 (((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦𝐵) → ∃𝑥 𝑥𝐹𝑦)
9 ffn 5046 . . . . . . . . . . 11 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
10 fnbr 5001 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
1110ex 108 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
129, 11syl 14 . . . . . . . . . 10 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦𝑥𝐴))
1312ancrd 309 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦 → (𝑥𝐴𝑥𝐹𝑦)))
1413eximdv 1760 . . . . . . . 8 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥𝐴𝑥𝐹𝑦)))
15 df-rex 2312 . . . . . . . 8 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
1614, 15syl6ibr 151 . . . . . . 7 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑥𝐹𝑦))
1716ad2antrr 457 . . . . . 6 (((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦𝐵) → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑥𝐹𝑦))
188, 17mpd 13 . . . . 5 (((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑥𝐹𝑦)
1918ralrimiva 2392 . . . 4 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) → ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦)
202, 19jca 290 . . 3 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
211, 20sylbi 114 . 2 (𝐹:𝐴onto𝐵 → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
22 fnbrfvb 5214 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
2322biimprd 147 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦))
24 eqcom 2042 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
2523, 24syl6ib 150 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑥𝐹𝑦𝑦 = (𝐹𝑥)))
269, 25sylan 267 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → (𝑥𝐹𝑦𝑦 = (𝐹𝑥)))
2726reximdva 2421 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥𝐴 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
2827ralimdv 2388 . . . 4 (𝐹:𝐴𝐵 → (∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
2928imdistani 419 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
30 dffo3 5314 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
3129, 30sylibr 137 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) → 𝐹:𝐴onto𝐵)
3221, 31impbii 117 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wex 1381  wcel 1393  wral 2306  wrex 2307   class class class wbr 3764  ran crn 4346   Fn wfn 4897  wf 4898  ontowfo 4900  cfv 4902
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910
This theorem is referenced by:  dffo5  5316
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