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Theorem fdmdifeqresdif 44219
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 4721 . . . . 5 (𝑥 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑥 = 𝑌)
21adantl 482 . . . 4 ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝑥 ∈ (𝐷 ∖ {𝑌})) → ¬ 𝑥 = 𝑌)
32iffalsed 4480 . . 3 ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝑥 ∈ (𝐷 ∖ {𝑌})) → if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)) = (𝐺𝑥))
43mpteq2dva 5157 . 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 5847 . . 3 (𝐹 ↾ (𝐷 ∖ {𝑌})) = ((𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) ↾ (𝐷 ∖ {𝑌}))
7 difssd 4112 . . . 4 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐷 ∖ {𝑌}) ⊆ 𝐷)
87resmptd 5906 . . 3 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → ((𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))))
96, 8syl5eq 2872 . 2 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐹 ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))))
10 ffn 6510 . . 3 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 Fn (𝐷 ∖ {𝑌}))
11 dffn5 6720 . . 3 (𝐺 Fn (𝐷 ∖ {𝑌}) ↔ 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺𝑥)))
1210, 11sylib 219 . 2 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺𝑥)))
134, 9, 123eqtr4rd 2871 1 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wcel 2106  cdif 3936  ifcif 4469  {csn 4563  cmpt 5142  cres 5555   Fn wfn 6346  wf 6347  cfv 6351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-res 5565  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359
This theorem is referenced by:  lincext2  44339  lincext3  44340
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