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Theorem fconstfvm 5304
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5303. (Contributed by Jim Kingdon, 8-Jan-2019.)
Assertion
Ref Expression
fconstfvm (y y A → (𝐹:A⟶{B} ↔ (𝐹 Fn A x A (𝐹x) = B)))
Distinct variable groups:   x,A   x,B   x,𝐹   y,A
Allowed substitution hints:   B(y)   𝐹(y)

Proof of Theorem fconstfvm
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 4972 . . 3 (𝐹:A⟶{B} → 𝐹 Fn A)
2 fvconst 5276 . . . 4 ((𝐹:A⟶{B} x A) → (𝐹x) = B)
32ralrimiva 2370 . . 3 (𝐹:A⟶{B} → x A (𝐹x) = B)
41, 3jca 290 . 2 (𝐹:A⟶{B} → (𝐹 Fn A x A (𝐹x) = B))
5 fvelrnb 5146 . . . . . . . . 9 (𝐹 Fn A → (w ran 𝐹z A (𝐹z) = w))
6 fveq2 5103 . . . . . . . . . . . . . 14 (x = z → (𝐹x) = (𝐹z))
76eqeq1d 2030 . . . . . . . . . . . . 13 (x = z → ((𝐹x) = B ↔ (𝐹z) = B))
87rspccva 2632 . . . . . . . . . . . 12 ((x A (𝐹x) = B z A) → (𝐹z) = B)
98eqeq1d 2030 . . . . . . . . . . 11 ((x A (𝐹x) = B z A) → ((𝐹z) = wB = w))
109rexbidva 2301 . . . . . . . . . 10 (x A (𝐹x) = B → (z A (𝐹z) = wz A B = w))
11 r19.9rmv 3291 . . . . . . . . . . 11 (y y A → (B = wz A B = w))
1211bicomd 129 . . . . . . . . . 10 (y y A → (z A B = wB = w))
1310, 12sylan9bbr 439 . . . . . . . . 9 ((y y A x A (𝐹x) = B) → (z A (𝐹z) = wB = w))
145, 13sylan9bbr 439 . . . . . . . 8 (((y y A x A (𝐹x) = B) 𝐹 Fn A) → (w ran 𝐹B = w))
15 elsn 3365 . . . . . . . . 9 (w {B} ↔ w = B)
16 eqcom 2024 . . . . . . . . 9 (w = BB = w)
1715, 16bitr2i 174 . . . . . . . 8 (B = ww {B})
1814, 17syl6bb 185 . . . . . . 7 (((y y A x A (𝐹x) = B) 𝐹 Fn A) → (w ran 𝐹w {B}))
1918eqrdv 2020 . . . . . 6 (((y y A x A (𝐹x) = B) 𝐹 Fn A) → ran 𝐹 = {B})
2019an32s 490 . . . . 5 (((y y A 𝐹 Fn A) x A (𝐹x) = B) → ran 𝐹 = {B})
2120exp31 346 . . . 4 (y y A → (𝐹 Fn A → (x A (𝐹x) = B → ran 𝐹 = {B})))
2221imdistand 424 . . 3 (y y A → ((𝐹 Fn A x A (𝐹x) = B) → (𝐹 Fn A ran 𝐹 = {B})))
23 df-fo 4835 . . . 4 (𝐹:Aonto→{B} ↔ (𝐹 Fn A ran 𝐹 = {B}))
24 fof 5031 . . . 4 (𝐹:Aonto→{B} → 𝐹:A⟶{B})
2523, 24sylbir 125 . . 3 ((𝐹 Fn A ran 𝐹 = {B}) → 𝐹:A⟶{B})
2622, 25syl6 29 . 2 (y y A → ((𝐹 Fn A x A (𝐹x) = B) → 𝐹:A⟶{B}))
274, 26impbid2 131 1 (y y A → (𝐹:A⟶{B} ↔ (𝐹 Fn A x A (𝐹x) = B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  wral 2284  wrex 2285  {csn 3350  ran crn 4273   Fn wfn 4824  wf 4825  ontowfo 4827  cfv 4829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837
This theorem is referenced by:  fconst3m  5305
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