Theorem fdmdifeqresdif 47328

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 4790	. . . . 5 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑥 = 𝑌)
2	1	adantl 481	. . . 4 ⊢ ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅 ∧ 𝑥 ∈ (𝐷 ∖ {𝑌})) → ¬ 𝑥 = 𝑌)
3	2	iffalsed 4535	. . 3 ⊢ ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅 ∧ 𝑥 ∈ (𝐷 ∖ {𝑌})) → if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥)) = (𝐺‘𝑥))
4	3	mpteq2dva 5242	. 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 5975	. . 3 ⊢ (𝐹 ↾ (𝐷 ∖ {𝑌})) = ((𝑥 ∈ 𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))) ↾ (𝐷 ∖ {𝑌}))
7		difssd 4128	. . . 4 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐷 ∖ {𝑌}) ⊆ 𝐷)
8	7	resmptd 6038	. . 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 2779	. 2 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐹 ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))))
10		ffn 6716	. . 3 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 Fn (𝐷 ∖ {𝑌}))
11		dffn5 6951	. . 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 2778	1 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ∖ cdif 3941  ifcif 4524  {csn 4624   ↦ cmpt 5225   ↾ cres 5674   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550

This theorem is referenced by:  lincext2  47446  lincext3  47447