Theorem fconst4m 5324

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 5323	. 2 ⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ A ⊆ (◡𝐹 “ {B}))))
2		cnvimass 4631	. . . . . 6 ⊢ (◡𝐹 “ {B}) ⊆ dom 𝐹
3		fndm 4941	. . . . . 6 ⊢ (𝐹 Fn A → dom 𝐹 = A)
4	2, 3	syl5sseq 2987	. . . . 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 2954	. . . 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 427	. 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 1242  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911  {csn 3367  ^◡ccnv 4287  dom cdm 4288   “ cima 4291   Fn wfn 4840  ⟶wf 4841

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853

This theorem is referenced by: (None)