Theorem fconstfvm 5907

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5906. (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 5513	. . 3 ⊢ (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
2		fvconst 5877	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
3	2	ralrimiva 2617	. . 3 ⊢ (𝐹:𝐴⟶{𝐵} → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)
4	1, 3	jca 306	. 2 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
5		fvelrnb 5729	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤))
6		fveq2 5675	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
7	6	eqeq1d 2243	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝑧) = 𝐵))
8	7	rspccva 2922	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
9	8	eqeq1d 2243	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) = 𝑤 ↔ 𝐵 = 𝑤))
10	9	rexbidva 2541	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤 ↔ ∃𝑧 ∈ 𝐴 𝐵 = 𝑤))
11		r19.9rmv 3605	. . . . . . . . . . 11 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐵 = 𝑤 ↔ ∃𝑧 ∈ 𝐴 𝐵 = 𝑤))
12	11	bicomd 141	. . . . . . . . . 10 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∃𝑧 ∈ 𝐴 𝐵 = 𝑤 ↔ 𝐵 = 𝑤))
13	10, 12	sylan9bbr 463	. . . . . . . . 9 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → (∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤 ↔ 𝐵 = 𝑤))
14	5, 13	sylan9bbr 463	. . . . . . . 8 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹 ↔ 𝐵 = 𝑤))
15		velsn 3711	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝐵} ↔ 𝑤 = 𝐵)
16		eqcom 2236	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 ↔ 𝐵 = 𝑤)
17	15, 16	bitr2i 185	. . . . . . . 8 ⊢ (𝐵 = 𝑤 ↔ 𝑤 ∈ {𝐵})
18	14, 17	bitrdi 196	. . . . . . 7 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ {𝐵}))
19	18	eqrdv 2232	. . . . . 6 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ran 𝐹 = {𝐵})
20	19	an32s 570	. . . . 5 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ 𝐹 Fn 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → ran 𝐹 = {𝐵})
21	20	exp31 364	. . . 4 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐹 Fn 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → ran 𝐹 = {𝐵})))
22	21	imdistand 447	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵})))
23		df-fo 5363	. . . 4 ⊢ (𝐹:𝐴–onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
24		fof 5595	. . . 4 ⊢ (𝐹:𝐴–onto→{𝐵} → 𝐹:𝐴⟶{𝐵})
25	23, 24	sylbir 135	. . 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 143	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {csn 3694  ran crn 4755   Fn wfn 5352  ⟶wf 5353  –onto→wfo 5355  ‘cfv 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365

This theorem is referenced by:  fconst3m  5908