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Theorem fconstfvm 5322
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5321. (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 4989 . . 3 (𝐹:A⟶{B} → 𝐹 Fn A)
2 fvconst 5294 . . . 4 ((𝐹:A⟶{B} x A) → (𝐹x) = B)
32ralrimiva 2386 . . 3 (𝐹:A⟶{B} → x A (𝐹x) = B)
41, 3jca 290 . 2 (𝐹:A⟶{B} → (𝐹 Fn A x A (𝐹x) = B))
5 fvelrnb 5164 . . . . . . . . 9 (𝐹 Fn A → (w ran 𝐹z A (𝐹z) = w))
6 fveq2 5121 . . . . . . . . . . . . . 14 (x = z → (𝐹x) = (𝐹z))
76eqeq1d 2045 . . . . . . . . . . . . 13 (x = z → ((𝐹x) = B ↔ (𝐹z) = B))
87rspccva 2649 . . . . . . . . . . . 12 ((x A (𝐹x) = B z A) → (𝐹z) = B)
98eqeq1d 2045 . . . . . . . . . . 11 ((x A (𝐹x) = B z A) → ((𝐹z) = wB = w))
109rexbidva 2317 . . . . . . . . . 10 (x A (𝐹x) = B → (z A (𝐹z) = wz A B = w))
11 r19.9rmv 3307 . . . . . . . . . . 11 (y y A → (B = wz A B = w))
1211bicomd 129 . . . . . . . . . 10 (y y A → (z A B = wB = w))
1310, 12sylan9bbr 436 . . . . . . . . 9 ((y y A x A (𝐹x) = B) → (z A (𝐹z) = wB = w))
145, 13sylan9bbr 436 . . . . . . . 8 (((y y A x A (𝐹x) = B) 𝐹 Fn A) → (w ran 𝐹B = w))
15 elsn 3382 . . . . . . . . 9 (w {B} ↔ w = B)
16 eqcom 2039 . . . . . . . . 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 2035 . . . . . 6 (((y y A x A (𝐹x) = B) 𝐹 Fn A) → ran 𝐹 = {B})
2019an32s 502 . . . . 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 421 . . 3 (y y A → ((𝐹 Fn A x A (𝐹x) = B) → (𝐹 Fn A ran 𝐹 = {B})))
23 df-fo 4851 . . . 4 (𝐹:Aonto→{B} ↔ (𝐹 Fn A ran 𝐹 = {B}))
24 fof 5049 . . . 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 1242  wex 1378   wcel 1390  wral 2300  wrex 2301  {csn 3367  ran crn 4289   Fn wfn 4840  wf 4841  ontowfo 4843  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-bndl 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-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853
This theorem is referenced by:  fconst3m  5323
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