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Theorem fconst4 6625
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 6624 . 2 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
2 cnvimass 5625 . . . . . 6 (𝐹 “ {𝐵}) ⊆ dom 𝐹
3 fndm 6129 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
42, 3syl5sseq 3802 . . . . 5 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) ⊆ 𝐴)
54biantrurd 522 . . . 4 (𝐹 Fn 𝐴 → (𝐴 ⊆ (𝐹 “ {𝐵}) ↔ ((𝐹 “ {𝐵}) ⊆ 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
6 eqss 3767 . . . 4 ((𝐹 “ {𝐵}) = 𝐴 ↔ ((𝐹 “ {𝐵}) ⊆ 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
75, 6syl6bbr 278 . . 3 (𝐹 Fn 𝐴 → (𝐴 ⊆ (𝐹 “ {𝐵}) ↔ (𝐹 “ {𝐵}) = 𝐴))
87pm5.32i 564 . 2 ((𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})) ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))
91, 8bitri 264 1 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wss 3723  {csn 4317  ccnv 5249  dom cdm 5250  cima 5253   Fn wfn 6025  wf 6026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038
This theorem is referenced by:  lkr0f  34903
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