Theorem fdmdifeqresdif 47513

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 4791	. . . . 5 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑥 = 𝑌)
2	1	adantl 480	. . . 4 ⊢ ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅 ∧ 𝑥 ∈ (𝐷 ∖ {𝑌})) → ¬ 𝑥 = 𝑌)
3	2	iffalsed 4536	. . 3 ⊢ ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅 ∧ 𝑥 ∈ (𝐷 ∖ {𝑌})) → if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥)) = (𝐺‘𝑥))
4	3	mpteq2dva 5244	. 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 5976	. . 3 ⊢ (𝐹 ↾ (𝐷 ∖ {𝑌})) = ((𝑥 ∈ 𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))) ↾ (𝐷 ∖ {𝑌}))
7		difssd 4126	. . . 4 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐷 ∖ {𝑌}) ⊆ 𝐷)
8	7	resmptd 6040	. . 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	eqtrid 2777	. 2 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐹 ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))))
10		ffn 6717	. . 3 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 Fn (𝐷 ∖ {𝑌}))
11		dffn5 6950	. . 3 ⊢ (𝐺 Fn (𝐷 ∖ {𝑌}) ↔ 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺‘𝑥)))
12	10, 11	sylib 217	. 2 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺‘𝑥)))
13	4, 9, 12	3eqtr4rd 2776	1 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ∖ cdif 3938  ifcif 4525  {csn 4625   ↦ cmpt 5227   ↾ cres 5675   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-res 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551

This theorem is referenced by:  lincext2  47631  lincext3  47632