Theorem fconstfvm 5730

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5729. (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 5361	. . 3 ⊢ (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
2		fvconst 5700	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
3	2	ralrimiva 2550	. . 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 5559	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤))
6		fveq2 5511	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
7	6	eqeq1d 2186	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝑧) = 𝐵))
8	7	rspccva 2840	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
9	8	eqeq1d 2186	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) = 𝑤 ↔ 𝐵 = 𝑤))
10	9	rexbidva 2474	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤 ↔ ∃𝑧 ∈ 𝐴 𝐵 = 𝑤))
11		r19.9rmv 3514	. . . . . . . . . . 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 3608	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝐵} ↔ 𝑤 = 𝐵)
16		eqcom 2179	. . . . . . . . 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 2175	. . . . . 6 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ran 𝐹 = {𝐵})
20	19	an32s 568	. . . . 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 5218	. . . 4 ⊢ (𝐹:𝐴–onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
24		fof 5434	. . . 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 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {csn 3591  ran crn 4624   Fn wfn 5207  ⟶wf 5208  –onto→wfo 5210  ‘cfv 5212

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fo 5218  df-fv 5220

This theorem is referenced by:  fconst3m  5731