Theorem fconstfvm 5592

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5591. (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 5230	. . 3 ⊢ (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
2		fvconst 5562	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
3	2	ralrimiva 2479	. . 3 ⊢ (𝐹:𝐴⟶{𝐵} → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)
4	1, 3	jca 302	. 2 ⊢ (𝐹:𝐴⟶{𝐵} → (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
5		fvelrnb 5423	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤))
6		fveq2 5375	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
7	6	eqeq1d 2123	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝑧) = 𝐵))
8	7	rspccva 2759	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
9	8	eqeq1d 2123	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) = 𝑤 ↔ 𝐵 = 𝑤))
10	9	rexbidva 2408	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤 ↔ ∃𝑧 ∈ 𝐴 𝐵 = 𝑤))
11		r19.9rmv 3420	. . . . . . . . . . 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 456	. . . . . . . . 9 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → (∃𝑧 ∈ 𝐴 (𝐹‘𝑧) = 𝑤 ↔ 𝐵 = 𝑤))
14	5, 13	sylan9bbr 456	. . . . . . . 8 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑤 ∈ ran 𝐹 ↔ 𝐵 = 𝑤))
15		velsn 3510	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝐵} ↔ 𝑤 = 𝐵)
16		eqcom 2117	. . . . . . . . 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 2113	. . . . . 6 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ran 𝐹 = {𝐵})
20	19	an32s 540	. . . . 5 ⊢ (((∃𝑦 𝑦 ∈ 𝐴 ∧ 𝐹 Fn 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → ran 𝐹 = {𝐵})
21	20	exp31 359	. . . 4 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐹 Fn 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 → ran 𝐹 = {𝐵})))
22	21	imdistand 441	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵})))
23		df-fo 5087	. . . 4 ⊢ (𝐹:𝐴–onto→{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = {𝐵}))
24		fof 5303	. . . 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 1314  ∃wex 1451   ∈ wcel 1463  ∀wral 2390  ∃wrex 2391  {csn 3493  ran crn 4500   Fn wfn 5076  ⟶wf 5077  –onto→wfo 5079  ‘cfv 5081

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fo 5087  df-fv 5089

This theorem is referenced by:  fconst3m  5593