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Theorem fconst4 6973
 Description: Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
Assertion
Ref Expression
fconst4 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))

Proof of Theorem fconst4
StepHypRef Expression
1 fconst3 6972 . 2 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
2 cnvimass 5925 . . . . . 6 (𝐹 “ {𝐵}) ⊆ dom 𝐹
3 fndm 6440 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
42, 3sseqtrid 3946 . . . . 5 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) ⊆ 𝐴)
54biantrurd 536 . . . 4 (𝐹 Fn 𝐴 → (𝐴 ⊆ (𝐹 “ {𝐵}) ↔ ((𝐹 “ {𝐵}) ⊆ 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
6 eqss 3909 . . . 4 ((𝐹 “ {𝐵}) = 𝐴 ↔ ((𝐹 “ {𝐵}) ⊆ 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
75, 6bitr4di 292 . . 3 (𝐹 Fn 𝐴 → (𝐴 ⊆ (𝐹 “ {𝐵}) ↔ (𝐹 “ {𝐵}) = 𝐴))
87pm5.32i 578 . 2 ((𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})) ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))
91, 8bitri 278 1 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ⊆ wss 3860  {csn 4525  ◡ccnv 5526  dom cdm 5527   “ cima 5530   Fn wfn 6334  ⟶wf 6335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347 This theorem is referenced by:  lkr0f  36696
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