Theorem oprpiece1res2 24883

Description: Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)

Hypotheses

Ref	Expression
oprpiece1.1	⊢ 𝐴 ∈ ℝ
oprpiece1.2	⊢ 𝐵 ∈ ℝ
oprpiece1.3	⊢ 𝐴 ≤ 𝐵
oprpiece1.4	⊢ 𝑅 ∈ V
oprpiece1.5	⊢ 𝑆 ∈ V
oprpiece1.6	⊢ 𝐾 ∈ (𝐴[,]𝐵)
oprpiece1.7	⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))
oprpiece1.9	⊢ (𝑥 = 𝐾 → 𝑅 = 𝑃)
oprpiece1.10	⊢ (𝑥 = 𝐾 → 𝑆 = 𝑄)
oprpiece1.11	⊢ (𝑦 ∈ 𝐶 → 𝑃 = 𝑄)
oprpiece1.12	⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆)

Assertion

Ref	Expression
oprpiece1res2	⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐾,𝑦   𝑥,𝑃   𝑥,𝑄

Allowed substitution hints:   𝑃(𝑦)   𝑄(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem oprpiece1res2

Step	Hyp	Ref	Expression
1		oprpiece1.6	. . . 4 ⊢ 𝐾 ∈ (𝐴[,]𝐵)
2		oprpiece1.1	. . . . . 6 ⊢ 𝐴 ∈ ℝ
3	2	rexri 11208	. . . . 5 ⊢ 𝐴 ∈ ℝ*
4		oprpiece1.2	. . . . . 6 ⊢ 𝐵 ∈ ℝ
5	4	rexri 11208	. . . . 5 ⊢ 𝐵 ∈ ℝ*
6		oprpiece1.3	. . . . 5 ⊢ 𝐴 ≤ 𝐵
7		ubicc2 13402	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
8	3, 5, 6, 7	mp3an 1463	. . . 4 ⊢ 𝐵 ∈ (𝐴[,]𝐵)
9		iccss2 13354	. . . 4 ⊢ ((𝐾 ∈ (𝐴[,]𝐵) ∧ 𝐵 ∈ (𝐴[,]𝐵)) → (𝐾[,]𝐵) ⊆ (𝐴[,]𝐵))
10	1, 8, 9	mp2an 692	. . 3 ⊢ (𝐾[,]𝐵) ⊆ (𝐴[,]𝐵)
11		ssid 3966	. . 3 ⊢ 𝐶 ⊆ 𝐶
12		resmpo 7489	. . 3 ⊢ (((𝐾[,]𝐵) ⊆ (𝐴[,]𝐵) ∧ 𝐶 ⊆ 𝐶) → ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) ↾ ((𝐾[,]𝐵) × 𝐶)) = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)))
13	10, 11, 12	mp2an 692	. 2 ⊢ ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) ↾ ((𝐾[,]𝐵) × 𝐶)) = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))
14		oprpiece1.7	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))
15	14	reseq1i 5935	. 2 ⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = ((𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) ↾ ((𝐾[,]𝐵) × 𝐶))
16		oprpiece1.12	. . 3 ⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆)
17		oprpiece1.11	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝑃 = 𝑄)
18	17	ad2antlr 727	. . . . . 6 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑃 = 𝑄)
19		simpr 484	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 ≤ 𝐾)
20	2, 4	elicc2i 13349	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (𝐴[,]𝐵) ↔ (𝐾 ∈ ℝ ∧ 𝐴 ≤ 𝐾 ∧ 𝐾 ≤ 𝐵))
21	20	simp1bi 1145	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ (𝐴[,]𝐵) → 𝐾 ∈ ℝ)
22	1, 21	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐾 ∈ ℝ
23	22, 4	elicc2i 13349	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐾[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐾 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
24	23	simp2bi 1146	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐾[,]𝐵) → 𝐾 ≤ 𝑥)
25	24	ad2antrr 726	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝐾 ≤ 𝑥)
26	23	simp1bi 1145	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐾[,]𝐵) → 𝑥 ∈ ℝ)
27	26	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 ∈ ℝ)
28		letri3 11235	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (𝑥 = 𝐾 ↔ (𝑥 ≤ 𝐾 ∧ 𝐾 ≤ 𝑥)))
29	27, 22, 28	sylancl 586	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → (𝑥 = 𝐾 ↔ (𝑥 ≤ 𝐾 ∧ 𝐾 ≤ 𝑥)))
30	19, 25, 29	mpbir2and 713	. . . . . . 7 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑥 = 𝐾)
31		oprpiece1.9	. . . . . . 7 ⊢ (𝑥 = 𝐾 → 𝑅 = 𝑃)
32	30, 31	syl 17	. . . . . 6 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑅 = 𝑃)
33		oprpiece1.10	. . . . . . 7 ⊢ (𝑥 = 𝐾 → 𝑆 = 𝑄)
34	30, 33	syl 17	. . . . . 6 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑆 = 𝑄)
35	18, 32, 34	3eqtr4d 2774	. . . . 5 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ≤ 𝐾) → 𝑅 = 𝑆)
36		eqidd 2730	. . . . 5 ⊢ (((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) ∧ ¬ 𝑥 ≤ 𝐾) → 𝑆 = 𝑆)
37	35, 36	ifeqda 4521	. . . 4 ⊢ ((𝑥 ∈ (𝐾[,]𝐵) ∧ 𝑦 ∈ 𝐶) → if(𝑥 ≤ 𝐾, 𝑅, 𝑆) = 𝑆)
38	37	mpoeq3ia 7447	. . 3 ⊢ (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆)
39	16, 38	eqtr4i 2755	. 2 ⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆))
40	13, 15, 39	3eqtr4i 2762	1 ⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911  ifcif 4484   class class class wbr 5102   × cxp 5629   ↾ cres 5633  (class class class)co 7369   ∈ cmpo 7371  ℝcr 11043  ℝ^*cxr 11183   ≤ cle 11185  [,]cicc 13285

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-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-pre-lttri 11118  ax-pre-lttrn 11119

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3030  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-icc 13289

This theorem is referenced by: (None)