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Theorem fdmdifeqresdif 41891
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 eldifsn 4315 . . . . . 6 (𝑥 ∈ (𝐷 ∖ {𝑌}) ↔ (𝑥𝐷𝑥𝑌))
2 neneq 2799 . . . . . 6 (𝑥𝑌 → ¬ 𝑥 = 𝑌)
31, 2simplbiim 659 . . . . 5 (𝑥 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑥 = 𝑌)
43adantl 482 . . . 4 ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝑥 ∈ (𝐷 ∖ {𝑌})) → ¬ 𝑥 = 𝑌)
54iffalsed 4095 . . 3 ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝑥 ∈ (𝐷 ∖ {𝑌})) → if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)) = (𝐺𝑥))
65mpteq2dva 4742 . 2 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺𝑥)))
7 fdmdifeqresdif.f . . . 4 𝐹 = (𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)))
87reseq1i 5390 . . 3 (𝐹 ↾ (𝐷 ∖ {𝑌})) = ((𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) ↾ (𝐷 ∖ {𝑌}))
9 difssd 3736 . . . 4 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐷 ∖ {𝑌}) ⊆ 𝐷)
109resmptd 5450 . . 3 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → ((𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))))
118, 10syl5eq 2667 . 2 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐹 ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))))
12 ffn 6043 . . 3 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 Fn (𝐷 ∖ {𝑌}))
13 dffn5 6239 . . 3 (𝐺 Fn (𝐷 ∖ {𝑌}) ↔ 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺𝑥)))
1412, 13sylib 208 . 2 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺𝑥)))
156, 11, 143eqtr4rd 2666 1 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1482  wcel 1989  wne 2793  cdif 3569  ifcif 4084  {csn 4175  cmpt 4727  cres 5114   Fn wfn 5881  wf 5882  cfv 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-res 5124  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894
This theorem is referenced by:  lincext2  42015  lincext3  42016
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