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Theorem fconstfv 5456
 Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5454. (Contributed by NM, 27-Aug-2004.)
Assertion
Ref Expression
fconstfv (F:A–→{B} ↔ (F Fn A x A (Fx) = B))
Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem fconstfv
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 5223 . . 3 (F:A–→{B} → F Fn A)
2 fvconst 5440 . . . 4 ((F:A–→{B} x A) → (Fx) = B)
32ralrimiva 2697 . . 3 (F:A–→{B} → x A (Fx) = B)
41, 3jca 518 . 2 (F:A–→{B} → (F Fn A x A (Fx) = B))
5 fneq2 5174 . . . . . . 7 (A = → (F Fn AF Fn ))
6 fn0 5202 . . . . . . 7 (F Fn F = )
75, 6syl6bb 252 . . . . . 6 (A = → (F Fn AF = ))
8 f0 5248 . . . . . . 7 :–→{B}
9 feq1 5210 . . . . . . 7 (F = → (F:–→{B} ↔ :–→{B}))
108, 9mpbiri 224 . . . . . 6 (F = F:–→{B})
117, 10syl6bi 219 . . . . 5 (A = → (F Fn AF:–→{B}))
12 feq2 5211 . . . . 5 (A = → (F:A–→{B} ↔ F:–→{B}))
1311, 12sylibrd 225 . . . 4 (A = → (F Fn AF:A–→{B}))
1413adantrd 454 . . 3 (A = → ((F Fn A x A (Fx) = B) → F:A–→{B}))
15 fvelrnb 5365 . . . . . . . . . 10 (F Fn A → (y ran Fz A (Fz) = y))
16 fveq2 5328 . . . . . . . . . . . . . . 15 (x = z → (Fx) = (Fz))
1716eqeq1d 2361 . . . . . . . . . . . . . 14 (x = z → ((Fx) = B ↔ (Fz) = B))
1817rspccva 2954 . . . . . . . . . . . . 13 ((x A (Fx) = B z A) → (Fz) = B)
1918eqeq1d 2361 . . . . . . . . . . . 12 ((x A (Fx) = B z A) → ((Fz) = yB = y))
2019rexbidva 2631 . . . . . . . . . . 11 (x A (Fx) = B → (z A (Fz) = yz A B = y))
21 r19.9rzv 3644 . . . . . . . . . . . 12 (A → (B = yz A B = y))
2221bicomd 192 . . . . . . . . . . 11 (A → (z A B = yB = y))
2320, 22sylan9bbr 681 . . . . . . . . . 10 ((A x A (Fx) = B) → (z A (Fz) = yB = y))
2415, 23sylan9bbr 681 . . . . . . . . 9 (((A x A (Fx) = B) F Fn A) → (y ran FB = y))
25 elsn 3748 . . . . . . . . . 10 (y {B} ↔ y = B)
26 eqcom 2355 . . . . . . . . . 10 (y = BB = y)
2725, 26bitr2i 241 . . . . . . . . 9 (B = yy {B})
2824, 27syl6bb 252 . . . . . . . 8 (((A x A (Fx) = B) F Fn A) → (y ran Fy {B}))
2928eqrdv 2351 . . . . . . 7 (((A x A (Fx) = B) F Fn A) → ran F = {B})
3029an32s 779 . . . . . 6 (((A F Fn A) x A (Fx) = B) → ran F = {B})
3130exp31 587 . . . . 5 (A → (F Fn A → (x A (Fx) = B → ran F = {B})))
3231imdistand 673 . . . 4 (A → ((F Fn A x A (Fx) = B) → (F Fn A ran F = {B})))
33 df-fo 4793 . . . . 5 (F:Aonto→{B} ↔ (F Fn A ran F = {B}))
34 fof 5269 . . . . 5 (F:Aonto→{B} → F:A–→{B})
3533, 34sylbir 204 . . . 4 ((F Fn A ran F = {B}) → F:A–→{B})
3632, 35syl6 29 . . 3 (A → ((F Fn A x A (Fx) = B) → F:A–→{B}))
3714, 36pm2.61ine 2592 . 2 ((F Fn A x A (Fx) = B) → F:A–→{B})
384, 37impbii 180 1 (F:A–→{B} ↔ (F Fn A x A (Fx) = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  ∃wrex 2615  ∅c0 3550  {csn 3737  ran crn 4773   Fn wfn 4776  –→wf 4777  –onto→wfo 4779   ‘cfv 4781 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795 This theorem is referenced by:  fconst3  5457
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