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Theorem dffo3 5342
Description: An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
Assertion
Ref Expression
dffo3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dffo3
StepHypRef Expression
1 dffo2 5138 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
2 ffn 5074 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrnfv 5248 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
43eqeq1d 2064 . . . . 5 (𝐹 Fn 𝐴 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵))
52, 4syl 14 . . . 4 (𝐹:𝐴𝐵 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵))
6 simpr 107 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
7 ffvelrn 5328 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
87adantr 265 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) ∈ 𝐵)
96, 8eqeltrd 2130 . . . . . . . . . 10 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦𝐵)
109exp31 350 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥𝐴 → (𝑦 = (𝐹𝑥) → 𝑦𝐵)))
1110rexlimdv 2449 . . . . . . . 8 (𝐹:𝐴𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵))
1211biantrurd 293 . . . . . . 7 (𝐹:𝐴𝐵 → ((𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))))
13 dfbi2 374 . . . . . . 7 ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
1412, 13syl6rbbr 192 . . . . . 6 (𝐹:𝐴𝐵 → ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
1514albidv 1721 . . . . 5 (𝐹:𝐴𝐵 → (∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
16 abeq1 2163 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵))
17 df-ral 2328 . . . . 5 (∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
1815, 16, 173bitr4g 216 . . . 4 (𝐹:𝐴𝐵 → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
195, 18bitrd 181 . . 3 (𝐹:𝐴𝐵 → (ran 𝐹 = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
2019pm5.32i 435 . 2 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
211, 20bitri 177 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  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:  dffo4  5343  foelrn  5345  foco2  5346  fcofo  5452  foov  5675  cnref1o  8680
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