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Theorem fnressn 5438
 Description: A function restricted to a singleton. (Contributed by set.mm contributors, 9-Oct-2004.)
Assertion
Ref Expression
fnressn ((F Fn A B A) → (F {B}) = {B, (FB)})

Proof of Theorem fnressn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sneq 3744 . . . . . 6 (x = B → {x} = {B})
21reseq2d 4934 . . . . 5 (x = B → (F {x}) = (F {B}))
3 fveq2 5328 . . . . . . 7 (x = B → (Fx) = (FB))
4 opeq12 4580 . . . . . . 7 ((x = B (Fx) = (FB)) → x, (Fx) = B, (FB))
53, 4mpdan 649 . . . . . 6 (x = Bx, (Fx) = B, (FB))
65sneqd 3746 . . . . 5 (x = B → {x, (Fx)} = {B, (FB)})
72, 6eqeq12d 2367 . . . 4 (x = B → ((F {x}) = {x, (Fx)} ↔ (F {B}) = {B, (FB)}))
87imbi2d 307 . . 3 (x = B → ((F Fn A → (F {x}) = {x, (Fx)}) ↔ (F Fn A → (F {B}) = {B, (FB)})))
9 vex 2862 . . . . . . 7 x V
109snss 3838 . . . . . 6 (x A ↔ {x} A)
11 fnssres 5196 . . . . . 6 ((F Fn A {x} A) → (F {x}) Fn {x})
1210, 11sylan2b 461 . . . . 5 ((F Fn A x A) → (F {x}) Fn {x})
13 dffn2 5224 . . . . . 6 ((F {x}) Fn {x} ↔ (F {x}):{x}–→V)
149fsn2 5434 . . . . . 6 ((F {x}):{x}–→V ↔ (((F {x}) ‘x) V (F {x}) = {x, ((F {x}) ‘x)}))
15 fvex 5339 . . . . . . . 8 ((F {x}) ‘x) V
1615biantrur 492 . . . . . . 7 ((F {x}) = {x, ((F {x}) ‘x)} ↔ (((F {x}) ‘x) V (F {x}) = {x, ((F {x}) ‘x)}))
179snid 3760 . . . . . . . . . . 11 x {x}
18 fvres 5342 . . . . . . . . . . 11 (x {x} → ((F {x}) ‘x) = (Fx))
1917, 18ax-mp 8 . . . . . . . . . 10 ((F {x}) ‘x) = (Fx)
2019opeq2i 4582 . . . . . . . . 9 x, ((F {x}) ‘x) = x, (Fx)
2120sneqi 3745 . . . . . . . 8 {x, ((F {x}) ‘x)} = {x, (Fx)}
2221eqeq2i 2363 . . . . . . 7 ((F {x}) = {x, ((F {x}) ‘x)} ↔ (F {x}) = {x, (Fx)})
2316, 22bitr3i 242 . . . . . 6 ((((F {x}) ‘x) V (F {x}) = {x, ((F {x}) ‘x)}) ↔ (F {x}) = {x, (Fx)})
2413, 14, 233bitri 262 . . . . 5 ((F {x}) Fn {x} ↔ (F {x}) = {x, (Fx)})
2512, 24sylib 188 . . . 4 ((F Fn A x A) → (F {x}) = {x, (Fx)})
2625expcom 424 . . 3 (x A → (F Fn A → (F {x}) = {x, (Fx)}))
278, 26vtoclga 2920 . 2 (B A → (F Fn A → (F {B}) = {B, (FB)}))
2827impcom 419 1 ((F Fn A B A) → (F {B}) = {B, (FB)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  {csn 3737  ⟨cop 4561   ↾ cres 4774   Fn wfn 4776  –→wf 4777   ‘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-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795 This theorem is referenced by:  fressnfv  5439
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