Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fdmdifeqresdif Structured version   Visualization version   GIF version

Theorem fdmdifeqresdif 48187
Description: The restriction of a conditional mapping to function values of a function having a domain which is a difference with a singleton equals this function. (Contributed by AV, 23-Apr-2019.)
Hypothesis
Ref Expression
fdmdifeqresdif.f 𝐹 = (𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)))
Assertion
Ref Expression
fdmdifeqresdif (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))
Distinct variable groups:   𝑥,𝐷   𝑥,𝐺   𝑥,𝑅   𝑥,𝑌
Allowed substitution hints:   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem fdmdifeqresdif
StepHypRef Expression
1 eldifsnneq 4796 . . . . 5 (𝑥 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑥 = 𝑌)
21adantl 481 . . . 4 ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝑥 ∈ (𝐷 ∖ {𝑌})) → ¬ 𝑥 = 𝑌)
32iffalsed 4542 . . 3 ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝑥 ∈ (𝐷 ∖ {𝑌})) → if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)) = (𝐺𝑥))
43mpteq2dva 5248 . 2 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺𝑥)))
5 fdmdifeqresdif.f . . . 4 𝐹 = (𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)))
65reseq1i 5996 . . 3 (𝐹 ↾ (𝐷 ∖ {𝑌})) = ((𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) ↾ (𝐷 ∖ {𝑌}))
7 difssd 4147 . . . 4 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐷 ∖ {𝑌}) ⊆ 𝐷)
87resmptd 6060 . . 3 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → ((𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))))
96, 8eqtrid 2787 . 2 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐹 ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))))
10 ffn 6737 . . 3 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 Fn (𝐷 ∖ {𝑌}))
11 dffn5 6967 . . 3 (𝐺 Fn (𝐷 ∖ {𝑌}) ↔ 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺𝑥)))
1210, 11sylib 218 . 2 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺𝑥)))
134, 9, 123eqtr4rd 2786 1 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  cdif 3960  ifcif 4531  {csn 4631  cmpt 5231  cres 5691   Fn wfn 6558  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  lincext2  48301  lincext3  48302
  Copyright terms: Public domain W3C validator