Theorem dffo3f 7140

Description: An onto mapping expressed in terms of function values. As dffo3 7136 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
dffo3f.1	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
dffo3f	⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))

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

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem dffo3f

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffo2 6838	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
2		ffn 6747	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3		fnrnfv 6981	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑤 ∈ 𝐴 𝑦 = (𝐹‘𝑤)})
4		dffo3f.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
5		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
6	4, 5	nffv 6930	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤)
7	6	nfeq2 2926	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 = (𝐹‘𝑤)
8		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑤 𝑦 = (𝐹‘𝑥)
9		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
10	9	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑦 = (𝐹‘𝑤) ↔ 𝑦 = (𝐹‘𝑥)))
11	7, 8, 10	cbvrexw 3313	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝐴 𝑦 = (𝐹‘𝑤) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
12	11	abbii 2812	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑤 ∈ 𝐴 𝑦 = (𝐹‘𝑤)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
13	3, 12	eqtrdi 2796	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
14	13	eqeq1d 2742	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵))
15	2, 14	syl 17	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵))
16		dfbi2 474	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))))
17		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
18		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐵
19	4, 17, 18	nff 6743	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵
20		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
21		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = (𝐹‘𝑥))
22		ffvelcdm 7115	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
23	22	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ 𝐵)
24	21, 23	eqeltrd 2844	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 ∈ 𝐵)
25	24	exp31 419	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵)))
26	19, 20, 25	rexlimd 3272	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵))
27	26	biantrurd 532	. . . . . . 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)))))
28	16, 27	bitr4id 290	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))))
29	28	albidv 1919	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))))
30		eqabcb 2886	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵 ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵))
31		df-ral 3068	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
32	29, 30, 31	3bitr4g 314	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
33	15, 32	bitrd 279	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (ran 𝐹 = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
34	33	pm5.32i 574	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
35	1, 34	bitri 275	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2108  {cab 2717  Ⅎwnfc 2893  ∀wral 3067  ∃wrex 3076  ran crn 5701   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581

This theorem is referenced by:  foelrnf  7142  fompt  7152