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Theorem fdmdifeqresdif 46507
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 4755 . . . . 5 (𝑥 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑥 = 𝑌)
21adantl 483 . . . 4 ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝑥 ∈ (𝐷 ∖ {𝑌})) → ¬ 𝑥 = 𝑌)
32iffalsed 4501 . . 3 ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝑥 ∈ (𝐷 ∖ {𝑌})) → if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)) = (𝐺𝑥))
43mpteq2dva 5209 . 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 5937 . . 3 (𝐹 ↾ (𝐷 ∖ {𝑌})) = ((𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) ↾ (𝐷 ∖ {𝑌}))
7 difssd 4096 . . . 4 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐷 ∖ {𝑌}) ⊆ 𝐷)
87resmptd 5998 . . 3 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → ((𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))) ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))))
96, 8eqtrid 2785 . 2 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐹 ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥))))
10 ffn 6672 . . 3 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 Fn (𝐷 ∖ {𝑌}))
11 dffn5 6905 . . 3 (𝐺 Fn (𝐷 ∖ {𝑌}) ↔ 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺𝑥)))
1210, 11sylib 217 . 2 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺𝑥)))
134, 9, 123eqtr4rd 2784 1 (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  cdif 3911  ifcif 4490  {csn 4590  cmpt 5192  cres 5639   Fn wfn 6495  wf 6496  cfv 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-res 5649  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508
This theorem is referenced by:  lincext2  46626  lincext3  46627
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