Theorem dffo3 5824

Description: An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)

Assertion

Ref	Expression
dffo3	⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dffo3

Step	Hyp	Ref	Expression
1		dffo2 5594	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
2		ffn 5508	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3		fnrnfv 5723	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
4	3	eqeq1d 2241	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵))
5	2, 4	syl 14	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵))
6		dfbi2 388	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))))
7		simpr 110	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = (𝐹‘𝑥))
8		ffvelcdm 5810	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
9	8	adantr 276	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ 𝐵)
10	7, 9	eqeltrd 2309	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 ∈ 𝐵)
11	10	exp31 364	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵)))
12	11	rexlimdv 2659	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵))
13	12	biantrurd 305	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → ((𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) ↔ ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))))
14	6, 13	bitr4id 199	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))))
15	14	albidv 1873	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))))
16		abeq1 2342	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵 ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵))
17		df-ral 2525	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
18	15, 16, 17	3bitr4g 223	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
19	5, 18	bitrd 188	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (ran 𝐹 = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
20	19	pm5.32i 454	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
21	1, 20	bitri 184	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2203  {cab 2218  ∀wral 2520  ∃wrex 2521  ran crn 4750   Fn wfn 5347  ⟶wf 5348  –onto→wfo 5350  ‘cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360

This theorem is referenced by:  dffo4  5825  foco2  5926  fcofo  5957  foov  6201  0ct  7398  ctmlemr  7399  ctm  7400  ctssdclemn0  7401  ctssdccl  7402  enumctlemm  7405  cnref1o  9983  nninfctlemfo  12736  1arith  13065  ctiunctlemfo  13190  znf1o  14799  ioocosf1o  15719  mpodvdsmulf1o  15858