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Theorem fconst3m 5730
Description: Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
Assertion
Ref Expression
fconst3m (∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fconst3m
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fconstfvm 5729 . 2 (∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = 𝐵)))
2 fnfun 5308 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
3 fndm 5310 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4 eqimss2 3210 . . . . 5 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
53, 4syl 14 . . . 4 (𝐹 Fn 𝐴𝐴 ⊆ dom 𝐹)
6 funconstss 5629 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑦𝐴 (𝐹𝑦) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
72, 5, 6syl2anc 411 . . 3 (𝐹 Fn 𝐴 → (∀𝑦𝐴 (𝐹𝑦) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
87pm5.32i 454 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = 𝐵) ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
91, 8bitrdi 196 1 (∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  wral 2455  wss 3129  {csn 3591  ccnv 4621  dom cdm 4622  cima 4625  Fun wfun 5205   Fn wfn 5206  wf 5207  cfv 5211
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fo 5217  df-fv 5219
This theorem is referenced by:  fconst4m  5731
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