Theorem fconst4m 5306

Description: Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)

Assertion

Ref	Expression
fconst4m	⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ (◡𝐹 “ {B}) = A)))

Distinct variable group:   x,A

Allowed substitution hints:   B(x)   𝐹(x)

Proof of Theorem fconst4m

Step	Hyp	Ref	Expression
1		fconst3m 5305	. 2 ⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ A ⊆ (◡𝐹 “ {B}))))
2		cnvimass 4615	. . . . . 6 ⊢ (◡𝐹 “ {B}) ⊆ dom 𝐹
3		fndm 4924	. . . . . 6 ⊢ (𝐹 Fn A → dom 𝐹 = A)
4	2, 3	syl5sseq 2970	. . . . 5 ⊢ (𝐹 Fn A → (◡𝐹 “ {B}) ⊆ A)
5	4	biantrurd 289	. . . 4 ⊢ (𝐹 Fn A → (A ⊆ (◡𝐹 “ {B}) ↔ ((◡𝐹 “ {B}) ⊆ A ∧ A ⊆ (◡𝐹 “ {B}))))
6		eqss 2937	. . . 4 ⊢ ((◡𝐹 “ {B}) = A ↔ ((◡𝐹 “ {B}) ⊆ A ∧ A ⊆ (◡𝐹 “ {B})))
7	5, 6	syl6bbr 187	. . 3 ⊢ (𝐹 Fn A → (A ⊆ (◡𝐹 “ {B}) ↔ (◡𝐹 “ {B}) = A))
8	7	pm5.32i 430	. 2 ⊢ ((𝐹 Fn A ∧ A ⊆ (◡𝐹 “ {B})) ↔ (𝐹 Fn A ∧ (◡𝐹 “ {B}) = A))
9	1, 8	syl6bb 185	1 ⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ (◡𝐹 “ {B}) = A)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374   ⊆ wss 2894  {csn 3350  ^◡ccnv 4271  dom cdm 4272   “ cima 4275   Fn wfn 4824  ⟶wf 4825

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837

This theorem is referenced by: (None)