Theorem dffo3f 7091

Description: An onto mapping expressed in terms of function values. As dffo3 7087 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 6786	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
2		ffn 6695	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3		fnrnfv 6930	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑤 ∈ 𝐴 𝑦 = (𝐹‘𝑤)})
4		dffo3f.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
5		nfcv 2927	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
6	4, 5	nffv 6881	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤)
7	6	nfeq2 2944	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 = (𝐹‘𝑤)
8		nfv 1937	. . . . . . . . 9 ⊢ Ⅎ𝑤 𝑦 = (𝐹‘𝑥)
9		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
10	9	eqeq2d 2776	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑦 = (𝐹‘𝑤) ↔ 𝑦 = (𝐹‘𝑥)))
11	7, 8, 10	cbvrexw 3308	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝐴 𝑦 = (𝐹‘𝑤) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
12	11	abbii 2832	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑤 ∈ 𝐴 𝑦 = (𝐹‘𝑤)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
13	3, 12	eqtrdi 2816	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
14	13	eqeq1d 2767	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵))
15	2, 14	syl 18	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵))
16		dfbi2 479	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))))
17		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
18		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐵
19	4, 17, 18	nff 6691	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵
20		nfv 1937	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
21		simpr 489	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = (𝐹‘𝑥))
22		ffvelcdm 7066	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
23	22	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ 𝐵)
24	21, 23	eqeltrd 2865	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 ∈ 𝐵)
25	24	exp31 424	. . . . . . . . 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 541	. . . . . . 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 293	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))))
29	28	albidv 1943	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))))
30		eqabcb 2905	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵 ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵))
31		df-ral 3080	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∀𝑦(𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
32	29, 30, 31	3bitr4g 317	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
33	15, 32	bitrd 282	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (ran 𝐹 = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
34	33	pm5.32i 584	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
35	1, 34	bitri 278	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561   = wceq 1563   ∈ wcel 2145  {cab 2743  Ⅎwnfc 2912  ∀wral 3079  ∃wrex 3089  ran crn 5652   Fn wfn 6520  ⟶wf 6521  –onto→wfo 6523  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533

This theorem is referenced by:  foelrnf  7093  fompt  7103