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Theorem dffo4 5666
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 5444 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
2 simpl 109 . . . 4 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
3 vex 2742 . . . . . . . . . 10 𝑦 ∈ V
43elrn 4872 . . . . . . . . 9 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 𝑥𝐹𝑦)
5 eleq2 2241 . . . . . . . . 9 (ran 𝐹 = 𝐵 → (𝑦 ∈ ran 𝐹𝑦𝐵))
64, 5bitr3id 194 . . . . . . . 8 (ran 𝐹 = 𝐵 → (∃𝑥 𝑥𝐹𝑦𝑦𝐵))
76biimpar 297 . . . . . . 7 ((ran 𝐹 = 𝐵𝑦𝐵) → ∃𝑥 𝑥𝐹𝑦)
87adantll 476 . . . . . 6 (((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦𝐵) → ∃𝑥 𝑥𝐹𝑦)
9 ffn 5367 . . . . . . . . . . 11 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
10 fnbr 5320 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
1110ex 115 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
129, 11syl 14 . . . . . . . . . 10 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦𝑥𝐴))
1312ancrd 326 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦 → (𝑥𝐴𝑥𝐹𝑦)))
1413eximdv 1880 . . . . . . . 8 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥𝐴𝑥𝐹𝑦)))
15 df-rex 2461 . . . . . . . 8 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
1614, 15imbitrrdi 162 . . . . . . 7 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑥𝐹𝑦))
1716ad2antrr 488 . . . . . 6 (((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦𝐵) → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑥𝐹𝑦))
188, 17mpd 13 . . . . 5 (((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑥𝐹𝑦)
1918ralrimiva 2550 . . . 4 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) → ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦)
202, 19jca 306 . . 3 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
211, 20sylbi 121 . 2 (𝐹:𝐴onto𝐵 → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
22 fnbrfvb 5558 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
2322biimprd 158 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑥𝐹𝑦 → (𝐹𝑥) = 𝑦))
24 eqcom 2179 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
2523, 24imbitrdi 161 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑥𝐹𝑦𝑦 = (𝐹𝑥)))
269, 25sylan 283 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → (𝑥𝐹𝑦𝑦 = (𝐹𝑥)))
2726reximdva 2579 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥𝐴 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
2827ralimdv 2545 . . . 4 (𝐹:𝐴𝐵 → (∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
2928imdistani 445 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
30 dffo3 5665 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
3129, 30sylibr 134 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) → 𝐹:𝐴onto𝐵)
3221, 31impbii 126 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456   class class class wbr 4005  ran crn 4629   Fn wfn 5213  wf 5214  ontowfo 5216  cfv 5218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226
This theorem is referenced by:  dffo5  5667
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