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Theorem fconst3m 5323
Description: Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
Assertion
Ref Expression
fconst3m (x x A → (𝐹:A⟶{B} ↔ (𝐹 Fn A A ⊆ (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 5322 . 2 (x x A → (𝐹:A⟶{B} ↔ (𝐹 Fn A y A (𝐹y) = B)))
2 fnfun 4939 . . . 4 (𝐹 Fn A → Fun 𝐹)
3 fndm 4941 . . . . 5 (𝐹 Fn A → dom 𝐹 = A)
4 eqimss2 2992 . . . . 5 (dom 𝐹 = AA ⊆ dom 𝐹)
53, 4syl 14 . . . 4 (𝐹 Fn AA ⊆ dom 𝐹)
6 funconstss 5228 . . . 4 ((Fun 𝐹 A ⊆ dom 𝐹) → (y A (𝐹y) = BA ⊆ (𝐹 “ {B})))
72, 5, 6syl2anc 391 . . 3 (𝐹 Fn A → (y A (𝐹y) = BA ⊆ (𝐹 “ {B})))
87pm5.32i 427 . 2 ((𝐹 Fn A y A (𝐹y) = B) ↔ (𝐹 Fn A A ⊆ (𝐹 “ {B})))
91, 8syl6bb 185 1 (x x A → (𝐹:A⟶{B} ↔ (𝐹 Fn A A ⊆ (𝐹 “ {B}))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  wral 2300  wss 2911  {csn 3367  ccnv 4287  dom cdm 4288  cima 4291  Fun wfun 4839   Fn wfn 4840  wf 4841  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853
This theorem is referenced by:  fconst4m  5324
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