Theorem fconstfvm 5322

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5321. (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 4989	. . 3 ⊢ (𝐹:A⟶{B} → 𝐹 Fn A)
2		fvconst 5294	. . . 4 ⊢ ((𝐹:A⟶{B} ∧ x ∈ A) → (𝐹‘x) = B)
3	2	ralrimiva 2386	. . 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 5164	. . . . . . . . 9 ⊢ (𝐹 Fn A → (w ∈ ran 𝐹 ↔ ∃z ∈ A (𝐹‘z) = w))
6		fveq2 5121	. . . . . . . . . . . . . 14 ⊢ (x = z → (𝐹‘x) = (𝐹‘z))
7	6	eqeq1d 2045	. . . . . . . . . . . . 13 ⊢ (x = z → ((𝐹‘x) = B ↔ (𝐹‘z) = B))
8	7	rspccva 2649	. . . . . . . . . . . 12 ⊢ ((∀x ∈ A (𝐹‘x) = B ∧ z ∈ A) → (𝐹‘z) = B)
9	8	eqeq1d 2045	. . . . . . . . . . 11 ⊢ ((∀x ∈ A (𝐹‘x) = B ∧ z ∈ A) → ((𝐹‘z) = w ↔ B = w))
10	9	rexbidva 2317	. . . . . . . . . 10 ⊢ (∀x ∈ A (𝐹‘x) = B → (∃z ∈ A (𝐹‘z) = w ↔ ∃z ∈ A B = w))
11		r19.9rmv 3307	. . . . . . . . . . 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 436	. . . . . . . . 9 ⊢ ((∃y y ∈ A ∧ ∀x ∈ A (𝐹‘x) = B) → (∃z ∈ A (𝐹‘z) = w ↔ B = w))
14	5, 13	sylan9bbr 436	. . . . . . . 8 ⊢ (((∃y y ∈ A ∧ ∀x ∈ A (𝐹‘x) = B) ∧ 𝐹 Fn A) → (w ∈ ran 𝐹 ↔ B = w))
15		elsn 3382	. . . . . . . . 9 ⊢ (w ∈ {B} ↔ w = B)
16		eqcom 2039	. . . . . . . . 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 2035	. . . . . 6 ⊢ (((∃y y ∈ A ∧ ∀x ∈ A (𝐹‘x) = B) ∧ 𝐹 Fn A) → ran 𝐹 = {B})
20	19	an32s 502	. . . . 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 421	. . 3 ⊢ (∃y y ∈ A → ((𝐹 Fn A ∧ ∀x ∈ A (𝐹‘x) = B) → (𝐹 Fn A ∧ ran 𝐹 = {B})))
23		df-fo 4851	. . . 4 ⊢ (𝐹:A–onto→{B} ↔ (𝐹 Fn A ∧ ran 𝐹 = {B}))
24		fof 5049	. . . 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 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  {csn 3367  ran crn 4289   Fn wfn 4840  ⟶wf 4841  –onto→wfo 4843  ‘cfv 4845

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853

This theorem is referenced by:  fconst3m  5323