Theorem fconstfvm 5400

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5399. (Contributed by Jim Kingdon, 8-Jan-2019.)

Assertion

Ref	Expression
fconstfvm	⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)))

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

Allowed substitution hints:   𝐵(𝑦)   𝐹(𝑦)

Proof of Theorem fconstfvm

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5066	. . 3 ⊢ (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
2		fvconst 5372	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
3	2	ralrimiva 2434	. . 3 ⊢ (𝐹:𝐴⟶{𝐵} → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)
4	1, 3	jca 300	. 2 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
5		fvelrnb 5242	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤))
6		fveq2 5198	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
7	6	eqeq1d 2089	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝑧) = 𝐵))
8	7	rspccva 2700	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
9	8	eqeq1d 2089	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) = 𝑤 ↔ 𝐵 = 𝑤))
10	9	rexbidva 2365	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤 ↔ ∃𝑧 ∈ 𝐴 𝐵 = 𝑤))
11		r19.9rmv 3333	. . . . . . . . . . 11 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐵 = 𝑤 ↔ ∃𝑧 ∈ 𝐴 𝐵 = 𝑤))
12	11	bicomd 139	. . . . . . . . . 10 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∃𝑧 ∈ 𝐴 𝐵 = 𝑤 ↔ 𝐵 = 𝑤))
13	10, 12	sylan9bbr 450	. . . . . . . . 9 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → (∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤 ↔ 𝐵 = 𝑤))
14	5, 13	sylan9bbr 450	. . . . . . . 8 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹 ↔ 𝐵 = 𝑤))
15		velsn 3415	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝐵} ↔ 𝑤 = 𝐵)
16		eqcom 2083	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 ↔ 𝐵 = 𝑤)
17	15, 16	bitr2i 183	. . . . . . . 8 ⊢ (𝐵 = 𝑤 ↔ 𝑤 ∈ {𝐵})
18	14, 17	syl6bb 194	. . . . . . 7 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ {𝐵}))
19	18	eqrdv 2079	. . . . . 6 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ran 𝐹 = {𝐵})
20	19	an32s 532	. . . . 5 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ 𝐹 Fn 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → ran 𝐹 = {𝐵})
21	20	exp31 356	. . . 4 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐹 Fn 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → ran 𝐹 = {𝐵})))
22	21	imdistand 435	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵})))
23		df-fo 4928	. . . 4 ⊢ (𝐹:𝐴–onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
24		fof 5126	. . . 4 ⊢ (𝐹:𝐴–onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
25	23, 24	sylbir 133	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}) → 𝐹:𝐴⟶{𝐵})
26	22, 25	syl6 33	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → 𝐹:𝐴⟶{𝐵}))
27	4, 26	impbid2 141	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349  {csn 3398  ran crn 4364   Fn wfn 4917  ⟶wf 4918  –onto→wfo 4920  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930

This theorem is referenced by:  fconst3m  5401