Theorem fdmdifeqresdif 48303

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 4751	. . . . 5 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑥 = 𝑌)
2	1	adantl 481	. . . 4 ⊢ ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅 ∧ 𝑥 ∈ (𝐷 ∖ {𝑌})) → ¬ 𝑥 = 𝑌)
3	2	iffalsed 4495	. . 3 ⊢ ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅 ∧ 𝑥 ∈ (𝐷 ∖ {𝑌})) → if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥)) = (𝐺‘𝑥))
4	3	mpteq2dva 5195	. 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 5935	. . 3 ⊢ (𝐹 ↾ (𝐷 ∖ {𝑌})) = ((𝑥 ∈ 𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))) ↾ (𝐷 ∖ {𝑌}))
7		difssd 4096	. . . 4 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐷 ∖ {𝑌}) ⊆ 𝐷)
8	7	resmptd 6000	. . 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 2776	. 2 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐹 ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))))
10		ffn 6670	. . 3 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 Fn (𝐷 ∖ {𝑌}))
11		dffn5 6901	. . 3 ⊢ (𝐺 Fn (𝐷 ∖ {𝑌}) ↔ 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺‘𝑥)))
12	10, 11	sylib 218	. 2 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺‘𝑥)))
13	4, 9, 12	3eqtr4rd 2775	1 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3908  ifcif 4484  {csn 4585   ↦ cmpt 5183   ↾ cres 5633   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507

This theorem is referenced by:  lincext2  48417  lincext3  48418