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Theorem fconst4m 5757
Description: Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
Assertion
Ref Expression
fconst4m (∃𝑥 𝑥𝐴 → (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
StepHypRef Expression
1 fconst3m 5756 . 2 (∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
2 cnvimass 5009 . . . . . 6 (𝐹 “ {𝐵}) ⊆ dom 𝐹
3 fndm 5334 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
42, 3sseqtrid 3220 . . . . 5 (𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) ⊆ 𝐴)
54biantrurd 305 . . . 4 (𝐹 Fn 𝐴 → (𝐴 ⊆ (𝐹 “ {𝐵}) ↔ ((𝐹 “ {𝐵}) ⊆ 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
6 eqss 3185 . . . 4 ((𝐹 “ {𝐵}) = 𝐴 ↔ ((𝐹 “ {𝐵}) ⊆ 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
75, 6bitr4di 198 . . 3 (𝐹 Fn 𝐴 → (𝐴 ⊆ (𝐹 “ {𝐵}) ↔ (𝐹 “ {𝐵}) = 𝐴))
87pm5.32i 454 . 2 ((𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})) ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))
91, 8bitrdi 196 1 (∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  wss 3144  {csn 3607  ccnv 4643  dom cdm 4644  cima 4647   Fn wfn 5230  wf 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fo 5241  df-fv 5243
This theorem is referenced by: (None)
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