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Theorem fconst4 6834
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 6833 . 2 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
2 cnvimass 5817 . . . . . 6 (𝐹 “ {𝐵}) ⊆ dom 𝐹
3 fndm 6317 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
42, 3sseqtrid 3935 . . . . 5 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) ⊆ 𝐴)
54biantrurd 533 . . . 4 (𝐹 Fn 𝐴 → (𝐴 ⊆ (𝐹 “ {𝐵}) ↔ ((𝐹 “ {𝐵}) ⊆ 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
6 eqss 3899 . . . 4 ((𝐹 “ {𝐵}) = 𝐴 ↔ ((𝐹 “ {𝐵}) ⊆ 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
75, 6syl6bbr 290 . . 3 (𝐹 Fn 𝐴 → (𝐴 ⊆ (𝐹 “ {𝐵}) ↔ (𝐹 “ {𝐵}) = 𝐴))
87pm5.32i 575 . 2 ((𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})) ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))
91, 8bitri 276 1 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1520  wss 3854  {csn 4466  ccnv 5434  dom cdm 5435  cima 5438   Fn wfn 6212  wf 6213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-fv 6225
This theorem is referenced by:  lkr0f  35711
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