Theorem fconstfvm 5304

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5303. (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 4972	. . 3 ⊢ (𝐹:A⟶{B} → 𝐹 Fn A)
2		fvconst 5276	. . . 4 ⊢ ((𝐹:A⟶{B} ∧ x ∈ A) → (𝐹‘x) = B)
3	2	ralrimiva 2370	. . 3 ⊢ (𝐹:A⟶{B} → ∀x ∈ A (𝐹‘x) = B)
4	1, 3	jca 290	. 2 ⊢ (𝐹:A⟶{B} → (𝐹 Fn A ∧ ∀x ∈ A (𝐹‘x) = B))
5		fvelrnb 5146	. . . . . . . . 9 ⊢ (𝐹 Fn A → (w ∈ ran 𝐹 ↔ ∃z ∈ A (𝐹‘z) = w))
6		fveq2 5103	. . . . . . . . . . . . . 14 ⊢ (x = z → (𝐹‘x) = (𝐹‘z))
7	6	eqeq1d 2030	. . . . . . . . . . . . 13 ⊢ (x = z → ((𝐹‘x) = B ↔ (𝐹‘z) = B))
8	7	rspccva 2632	. . . . . . . . . . . 12 ⊢ ((∀x ∈ A (𝐹‘x) = B ∧ z ∈ A) → (𝐹‘z) = B)
9	8	eqeq1d 2030	. . . . . . . . . . 11 ⊢ ((∀x ∈ A (𝐹‘x) = B ∧ z ∈ A) → ((𝐹‘z) = w ↔ B = w))
10	9	rexbidva 2301	. . . . . . . . . 10 ⊢ (∀x ∈ A (𝐹‘x) = B → (∃z ∈ A (𝐹‘z) = w ↔ ∃z ∈ A B = w))
11		r19.9rmv 3291	. . . . . . . . . . 11 ⊢ (∃y y ∈ A → (B = w ↔ ∃z ∈ A B = w))
12	11	bicomd 129	. . . . . . . . . 10 ⊢ (∃y y ∈ A → (∃z ∈ A B = w ↔ B = w))
13	10, 12	sylan9bbr 439	. . . . . . . . 9 ⊢ ((∃y y ∈ A ∧ ∀x ∈ A (𝐹‘x) = B) → (∃z ∈ A (𝐹‘z) = w ↔ B = w))
14	5, 13	sylan9bbr 439	. . . . . . . 8 ⊢ (((∃y y ∈ A ∧ ∀x ∈ A (𝐹‘x) = B) ∧ 𝐹 Fn A) → (w ∈ ran 𝐹 ↔ B = w))
15		elsn 3365	. . . . . . . . 9 ⊢ (w ∈ {B} ↔ w = B)
16		eqcom 2024	. . . . . . . . 9 ⊢ (w = B ↔ B = w)
17	15, 16	bitr2i 174	. . . . . . . 8 ⊢ (B = w ↔ w ∈ {B})
18	14, 17	syl6bb 185	. . . . . . 7 ⊢ (((∃y y ∈ A ∧ ∀x ∈ A (𝐹‘x) = B) ∧ 𝐹 Fn A) → (w ∈ ran 𝐹 ↔ w ∈ {B}))
19	18	eqrdv 2020	. . . . . 6 ⊢ (((∃y y ∈ A ∧ ∀x ∈ A (𝐹‘x) = B) ∧ 𝐹 Fn A) → ran 𝐹 = {B})
20	19	an32s 490	. . . . 5 ⊢ (((∃y y ∈ A ∧ 𝐹 Fn A) ∧ ∀x ∈ A (𝐹‘x) = B) → ran 𝐹 = {B})
21	20	exp31 346	. . . 4 ⊢ (∃y y ∈ A → (𝐹 Fn A → (∀x ∈ A (𝐹‘x) = B → ran 𝐹 = {B})))
22	21	imdistand 424	. . 3 ⊢ (∃y y ∈ A → ((𝐹 Fn A ∧ ∀x ∈ A (𝐹‘x) = B) → (𝐹 Fn A ∧ ran 𝐹 = {B})))
23		df-fo 4835	. . . 4 ⊢ (𝐹:A–onto→{B} ↔ (𝐹 Fn A ∧ ran 𝐹 = {B}))
24		fof 5031	. . . 4 ⊢ (𝐹:A–onto→{B} → 𝐹:A⟶{B})
25	23, 24	sylbir 125	. . 3 ⊢ ((𝐹 Fn A ∧ ran 𝐹 = {B}) → 𝐹:A⟶{B})
26	22, 25	syl6 29	. 2 ⊢ (∃y y ∈ A → ((𝐹 Fn A ∧ ∀x ∈ A (𝐹‘x) = B) → 𝐹:A⟶{B}))
27	4, 26	impbid2 131	1 ⊢ (∃y y ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ ∀x ∈ A (𝐹‘x) = B)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∀wral 2284  ∃wrex 2285  {csn 3350  ran crn 4273   Fn wfn 4824  ⟶wf 4825  –onto→wfo 4827  ‘cfv 4829

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837

This theorem is referenced by:  fconst3m  5305