Theorem fconstfvm 5638

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5637. (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 5272	. . 3 ⊢ (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
2		fvconst 5608	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
3	2	ralrimiva 2505	. . 3 ⊢ (𝐹:𝐴⟶{𝐵} → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)
4	1, 3	jca 304	. 2 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
5		fvelrnb 5469	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤))
6		fveq2 5421	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
7	6	eqeq1d 2148	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝑧) = 𝐵))
8	7	rspccva 2788	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
9	8	eqeq1d 2148	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) = 𝑤 ↔ 𝐵 = 𝑤))
10	9	rexbidva 2434	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤 ↔ ∃𝑧 ∈ 𝐴 𝐵 = 𝑤))
11		r19.9rmv 3454	. . . . . . . . . . 11 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐵 = 𝑤 ↔ ∃𝑧 ∈ 𝐴 𝐵 = 𝑤))
12	11	bicomd 140	. . . . . . . . . 10 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∃𝑧 ∈ 𝐴 𝐵 = 𝑤 ↔ 𝐵 = 𝑤))
13	10, 12	sylan9bbr 458	. . . . . . . . 9 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → (∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤 ↔ 𝐵 = 𝑤))
14	5, 13	sylan9bbr 458	. . . . . . . 8 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹 ↔ 𝐵 = 𝑤))
15		velsn 3544	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝐵} ↔ 𝑤 = 𝐵)
16		eqcom 2141	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 ↔ 𝐵 = 𝑤)
17	15, 16	bitr2i 184	. . . . . . . 8 ⊢ (𝐵 = 𝑤 ↔ 𝑤 ∈ {𝐵})
18	14, 17	syl6bb 195	. . . . . . 7 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ {𝐵}))
19	18	eqrdv 2137	. . . . . 6 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ran 𝐹 = {𝐵})
20	19	an32s 557	. . . . 5 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ 𝐹 Fn 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → ran 𝐹 = {𝐵})
21	20	exp31 361	. . . 4 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐹 Fn 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → ran 𝐹 = {𝐵})))
22	21	imdistand 443	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵})))
23		df-fo 5129	. . . 4 ⊢ (𝐹:𝐴–onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
24		fof 5345	. . . 4 ⊢ (𝐹:𝐴–onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
25	23, 24	sylbir 134	. . 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 142	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {csn 3527  ran crn 4540   Fn wfn 5118  ⟶wf 5119  –onto→wfo 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:  fconst3m  5639