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 48258
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 4791 . . . . 5 (𝑥 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑥 = 𝑌)
21adantl 481 . . . 4 ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝑥 ∈ (𝐷 ∖ {𝑌})) → ¬ 𝑥 = 𝑌)
32iffalsed 4536 . . 3 ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝑥 ∈ (𝐷 ∖ {𝑌})) → if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)) = (𝐺𝑥))
43mpteq2dva 5242 . 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 5993 . . 3 (𝐹 ↾ (𝐷 ∖ {𝑌})) = ((𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) ↾ (𝐷 ∖ {𝑌}))
7 difssd 4137 . . . 4 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐷 ∖ {𝑌}) ⊆ 𝐷)
87resmptd 6058 . . 3 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → ((𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))))
96, 8eqtrid 2789 . 2 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐹 ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))))
10 ffn 6736 . . 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 2788 1 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  cdif 3948  ifcif 4525  {csn 4626  cmpt 5225  cres 5687   Fn wfn 6556  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  lincext2  48372  lincext3  48373
  Copyright terms: Public domain W3C validator