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Theorem dffo3 5567
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 5349 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
2 ffn 5272 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrnfv 5468 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
43eqeq1d 2148 . . . . 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 109 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
7 ffvelrn 5553 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
87adantr 274 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) ∈ 𝐵)
96, 8eqeltrd 2216 . . . . . . . . . 10 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦𝐵)
109exp31 361 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥𝐴 → (𝑦 = (𝐹𝑥) → 𝑦𝐵)))
1110rexlimdv 2548 . . . . . . . 8 (𝐹:𝐴𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵))
1211biantrurd 303 . . . . . . 7 (𝐹:𝐴𝐵 → ((𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))))
13 dfbi2 385 . . . . . . 7 ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
1412, 13syl6rbbr 198 . . . . . 6 (𝐹:𝐴𝐵 → ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
1514albidv 1796 . . . . 5 (𝐹:𝐴𝐵 → (∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
16 abeq1 2249 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵))
17 df-ral 2421 . . . . 5 (∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
1815, 16, 173bitr4g 222 . . . 4 (𝐹:𝐴𝐵 → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
195, 18bitrd 187 . . 3 (𝐹:𝐴𝐵 → (ran 𝐹 = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
2019pm5.32i 449 . 2 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
211, 20bitri 183 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329   = wceq 1331  wcel 1480  {cab 2125  wral 2416  wrex 2417  ran crn 4540   Fn wfn 5118  wf 5119  ontowfo 5121  cfv 5123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131
This theorem is referenced by:  dffo4  5568  foco2  5655  fcofo  5685  foov  5917  0ct  6992  ctmlemr  6993  ctm  6994  ctssdclemn0  6995  ctssdccl  6996  enumctlemm  6999  cnref1o  9454  ctiunctlemfo  11965  ioocosf1o  12959
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