Theorem fdmdifeqresdif 44397

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 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥)))

Assertion

Ref	Expression
fdmdifeqresdif	⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (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

Step	Hyp	Ref	Expression
1		eldifsnneq 4726	. . . . 5 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑥 = 𝑌)
2	1	adantl 484	. . . 4 ⊢ ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅 ∧ 𝑥 ∈ (𝐷 ∖ {𝑌})) → ¬ 𝑥 = 𝑌)
3	2	iffalsed 4481	. . 3 ⊢ ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅 ∧ 𝑥 ∈ (𝐷 ∖ {𝑌})) → if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥)) = (𝐺‘𝑥))
4	3	mpteq2dva 5164	. 2 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺‘𝑥)))
5		fdmdifeqresdif.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥)))
6	5	reseq1i 5852	. . 3 ⊢ (𝐹 ↾ (𝐷 ∖ {𝑌})) = ((𝑥 ∈ 𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))) ↾ (𝐷 ∖ {𝑌}))
7		difssd 4112	. . . 4 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐷 ∖ {𝑌}) ⊆ 𝐷)
8	7	resmptd 5911	. . 3 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → ((𝑥 ∈ 𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))) ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))))
9	6, 8	syl5eq 2871	. 2 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐹 ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))))
10		ffn 6517	. . 3 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 Fn (𝐷 ∖ {𝑌}))
11		dffn5 6727	. . 3 ⊢ (𝐺 Fn (𝐷 ∖ {𝑌}) ↔ 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺‘𝑥)))
12	10, 11	sylib 220	. 2 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺‘𝑥)))
13	4, 9, 12	3eqtr4rd 2870	1 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ∖ cdif 3936  ifcif 4470  {csn 4570   ↦ cmpt 5149   ↾ cres 5560   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-res 5570  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366

This theorem is referenced by:  lincext2  44517  lincext3  44518