Theorem fdmdifeqresdif 45240

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 4679	. . . . 5 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑌}) → ¬ 𝑥 = 𝑌)
2	1	adantl 485	. . . 4 ⊢ ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅 ∧ 𝑥 ∈ (𝐷 ∖ {𝑌})) → ¬ 𝑥 = 𝑌)
3	2	iffalsed 4425	. . 3 ⊢ ((𝐺:(𝐷 ∖ {𝑌})⟶𝑅 ∧ 𝑥 ∈ (𝐷 ∖ {𝑌})) → if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥)) = (𝐺‘𝑥))
4	3	mpteq2dva 5125	. 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 5821	. . 3 ⊢ (𝐹 ↾ (𝐷 ∖ {𝑌})) = ((𝑥 ∈ 𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))) ↾ (𝐷 ∖ {𝑌}))
7		difssd 4023	. . . 4 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐷 ∖ {𝑌}) ⊆ 𝐷)
8	7	resmptd 5882	. . 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 2785	. 2 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → (𝐹 ↾ (𝐷 ∖ {𝑌})) = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))))
10		ffn 6504	. . 3 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 Fn (𝐷 ∖ {𝑌}))
11		dffn5 6730	. . 3 ⊢ (𝐺 Fn (𝐷 ∖ {𝑌}) ↔ 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺‘𝑥)))
12	10, 11	sylib 221	. 2 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (𝑥 ∈ (𝐷 ∖ {𝑌}) ↦ (𝐺‘𝑥)))
13	4, 9, 12	3eqtr4rd 2784	1 ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2114   ∖ cdif 3840  ifcif 4414  {csn 4516   ↦ cmpt 5110   ↾ cres 5527   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-res 5537  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347

This theorem is referenced by:  lincext2  45359  lincext3  45360