Theorem fconstfv 5456

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5454. (Contributed by NM, 27-Aug-2004.)

Assertion

Ref	Expression
fconstfv	⊢ (F:A–→{B} ↔ (F Fn A ∧ ∀x ∈ A (F ‘x) = B))

Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem fconstfv

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5223	. . 3 ⊢ (F:A–→{B} → F Fn A)
2		fvconst 5440	. . . 4 ⊢ ((F:A–→{B} ∧ x ∈ A) → (F ‘x) = B)
3	2	ralrimiva 2697	. . 3 ⊢ (F:A–→{B} → ∀x ∈ A (F ‘x) = B)
4	1, 3	jca 518	. 2 ⊢ (F:A–→{B} → (F Fn A ∧ ∀x ∈ A (F ‘x) = B))
5		fneq2 5174	. . . . . . 7 ⊢ (A = ∅ → (F Fn A ↔ F Fn ∅))
6		fn0 5202	. . . . . . 7 ⊢ (F Fn ∅ ↔ F = ∅)
7	5, 6	syl6bb 252	. . . . . 6 ⊢ (A = ∅ → (F Fn A ↔ F = ∅))
8		f0 5248	. . . . . . 7 ⊢ ∅:∅–→{B}
9		feq1 5210	. . . . . . 7 ⊢ (F = ∅ → (F:∅–→{B} ↔ ∅:∅–→{B}))
10	8, 9	mpbiri 224	. . . . . 6 ⊢ (F = ∅ → F:∅–→{B})
11	7, 10	syl6bi 219	. . . . 5 ⊢ (A = ∅ → (F Fn A → F:∅–→{B}))
12		feq2 5211	. . . . 5 ⊢ (A = ∅ → (F:A–→{B} ↔ F:∅–→{B}))
13	11, 12	sylibrd 225	. . . 4 ⊢ (A = ∅ → (F Fn A → F:A–→{B}))
14	13	adantrd 454	. . 3 ⊢ (A = ∅ → ((F Fn A ∧ ∀x ∈ A (F ‘x) = B) → F:A–→{B}))
15		fvelrnb 5365	. . . . . . . . . 10 ⊢ (F Fn A → (y ∈ ran F ↔ ∃z ∈ A (F ‘z) = y))
16		fveq2 5328	. . . . . . . . . . . . . . 15 ⊢ (x = z → (F ‘x) = (F ‘z))
17	16	eqeq1d 2361	. . . . . . . . . . . . . 14 ⊢ (x = z → ((F ‘x) = B ↔ (F ‘z) = B))
18	17	rspccva 2954	. . . . . . . . . . . . 13 ⊢ ((∀x ∈ A (F ‘x) = B ∧ z ∈ A) → (F ‘z) = B)
19	18	eqeq1d 2361	. . . . . . . . . . . 12 ⊢ ((∀x ∈ A (F ‘x) = B ∧ z ∈ A) → ((F ‘z) = y ↔ B = y))
20	19	rexbidva 2631	. . . . . . . . . . 11 ⊢ (∀x ∈ A (F ‘x) = B → (∃z ∈ A (F ‘z) = y ↔ ∃z ∈ A B = y))
21		r19.9rzv 3644	. . . . . . . . . . . 12 ⊢ (A ≠ ∅ → (B = y ↔ ∃z ∈ A B = y))
22	21	bicomd 192	. . . . . . . . . . 11 ⊢ (A ≠ ∅ → (∃z ∈ A B = y ↔ B = y))
23	20, 22	sylan9bbr 681	. . . . . . . . . 10 ⊢ ((A ≠ ∅ ∧ ∀x ∈ A (F ‘x) = B) → (∃z ∈ A (F ‘z) = y ↔ B = y))
24	15, 23	sylan9bbr 681	. . . . . . . . 9 ⊢ (((A ≠ ∅ ∧ ∀x ∈ A (F ‘x) = B) ∧ F Fn A) → (y ∈ ran F ↔ B = y))
25		elsn 3748	. . . . . . . . . 10 ⊢ (y ∈ {B} ↔ y = B)
26		eqcom 2355	. . . . . . . . . 10 ⊢ (y = B ↔ B = y)
27	25, 26	bitr2i 241	. . . . . . . . 9 ⊢ (B = y ↔ y ∈ {B})
28	24, 27	syl6bb 252	. . . . . . . 8 ⊢ (((A ≠ ∅ ∧ ∀x ∈ A (F ‘x) = B) ∧ F Fn A) → (y ∈ ran F ↔ y ∈ {B}))
29	28	eqrdv 2351	. . . . . . 7 ⊢ (((A ≠ ∅ ∧ ∀x ∈ A (F ‘x) = B) ∧ F Fn A) → ran F = {B})
30	29	an32s 779	. . . . . 6 ⊢ (((A ≠ ∅ ∧ F Fn A) ∧ ∀x ∈ A (F ‘x) = B) → ran F = {B})
31	30	exp31 587	. . . . 5 ⊢ (A ≠ ∅ → (F Fn A → (∀x ∈ A (F ‘x) = B → ran F = {B})))
32	31	imdistand 673	. . . 4 ⊢ (A ≠ ∅ → ((F Fn A ∧ ∀x ∈ A (F ‘x) = B) → (F Fn A ∧ ran F = {B})))
33		df-fo 4793	. . . . 5 ⊢ (F:A–onto→{B} ↔ (F Fn A ∧ ran F = {B}))
34		fof 5269	. . . . 5 ⊢ (F:A–onto→{B} → F:A–→{B})
35	33, 34	sylbir 204	. . . 4 ⊢ ((F Fn A ∧ ran F = {B}) → F:A–→{B})
36	32, 35	syl6 29	. . 3 ⊢ (A ≠ ∅ → ((F Fn A ∧ ∀x ∈ A (F ‘x) = B) → F:A–→{B}))
37	14, 36	pm2.61ine 2592	. 2 ⊢ ((F Fn A ∧ ∀x ∈ A (F ‘x) = B) → F:A–→{B})
38	4, 37	impbii 180	1 ⊢ (F:A–→{B} ↔ (F Fn A ∧ ∀x ∈ A (F ‘x) = B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  ∃wrex 2615  ∅c0 3550  {csn 3737  ran crn 4773   Fn wfn 4776  –→wf 4777  –onto→wfo 4779   ‘cfv 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795

This theorem is referenced by:  fconst3  5457