Theorem fphpd 43000

Description: Pigeonhole principle expressed with implicit substitution. If the range is smaller than the domain, two inputs must be mapped to the same output. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)

Hypotheses

Ref	Expression
fphpd.a	⊢ (𝜑 → 𝐵 ≺ 𝐴)
fphpd.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
fphpd.c	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)

Assertion

Ref	Expression
fphpd	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem fphpd

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		domnsym 9029	. . . 4 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴)
2		fphpd.a	. . . 4 ⊢ (𝜑 → 𝐵 ≺ 𝐴)
3	1, 2	nsyl3 138	. . 3 ⊢ (𝜑 → ¬ 𝐴 ≼ 𝐵)
4		relsdom 8888	. . . . . . 7 ⊢ Rel ≺
5	4	brrelex1i 5678	. . . . . 6 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ∈ V)
6	2, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)) → 𝐵 ∈ V)
8		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ 𝐴)
9		nfcsb1v 3871	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
10	9	nfel1 2913	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 ∈ 𝐵
11	8, 10	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 ∈ 𝐵)
12		eleq1w 2817	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
13	12	anbi2d 630	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑎 ∈ 𝐴)))
14		csbeq1a 3861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
15	14	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝐶 ∈ 𝐵 ↔ ⦋𝑎 / 𝑥⦌𝐶 ∈ 𝐵))
16	13, 15	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 ∈ 𝐵)))
17		fphpd.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
18	11, 16, 17	chvarfv 2245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 ∈ 𝐵)
19	18	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝐴 → ⦋𝑎 / 𝑥⦌𝐶 ∈ 𝐵))
20	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)) → (𝑎 ∈ 𝐴 → ⦋𝑎 / 𝑥⦌𝐶 ∈ 𝐵))
21		csbid 3860	. . . . . . . . . . 11 ⊢ ⦋𝑥 / 𝑥⦌𝐶 = 𝐶
22		vex 3442	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
23		fphpd.c	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
24	22, 23	csbie 3882	. . . . . . . . . . 11 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐷
25	21, 24	eqeq12i 2752	. . . . . . . . . 10 ⊢ (⦋𝑥 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 ↔ 𝐶 = 𝐷)
26	25	imbi1i 349	. . . . . . . . 9 ⊢ ((⦋𝑥 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷 → 𝑥 = 𝑦))
27	26	2ralbii 3109	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (⦋𝑥 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦))
28		nfcsb1v 3871	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
29	9, 28	nfeq 2910	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶
30		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑎 = 𝑦
31	29, 30	nfim 1897	. . . . . . . . . 10 ⊢ Ⅎ𝑥(⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 → 𝑎 = 𝑦)
32		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑦(⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 → 𝑎 = 𝑏)
33		csbeq1 3850	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ⦋𝑥 / 𝑥⦌𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
34	33	eqeq1d 2736	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (⦋𝑥 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶))
35		equequ1 2026	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑥 = 𝑦 ↔ 𝑎 = 𝑦))
36	34, 35	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((⦋𝑥 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 → 𝑥 = 𝑦) ↔ (⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 → 𝑎 = 𝑦)))
37		csbeq1 3850	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶)
38	37	eqeq2d 2745	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶))
39		equequ2 2027	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑏))
40	38, 39	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → ((⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 → 𝑎 = 𝑦) ↔ (⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 → 𝑎 = 𝑏)))
41	31, 32, 36, 40	rspc2 3583	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (⦋𝑥 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 → 𝑥 = 𝑦) → (⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 → 𝑎 = 𝑏)))
42	41	com12 32	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (⦋𝑥 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 → 𝑥 = 𝑦) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 → 𝑎 = 𝑏)))
43	27, 42	sylbir 235	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 → 𝑎 = 𝑏)))
44		id 22	. . . . . . . 8 ⊢ ((⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 → 𝑎 = 𝑏) → (⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 → 𝑎 = 𝑏))
45		csbeq1 3850	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶)
46	44, 45	impbid1 225	. . . . . . 7 ⊢ ((⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 → 𝑎 = 𝑏) → (⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 ↔ 𝑎 = 𝑏))
47	43, 46	syl6 35	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 ↔ 𝑎 = 𝑏)))
48	47	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 ↔ 𝑎 = 𝑏)))
49	20, 48	dom2d 8928	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)) → (𝐵 ∈ V → 𝐴 ≼ 𝐵))
50	7, 49	mpd 15	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)) → 𝐴 ≼ 𝐵)
51	3, 50	mtand 815	. 2 ⊢ (𝜑 → ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦))
52		ancom 460	. . . . . . 7 ⊢ ((¬ 𝑥 = 𝑦 ∧ 𝐶 = 𝐷) ↔ (𝐶 = 𝐷 ∧ ¬ 𝑥 = 𝑦))
53		df-ne 2931	. . . . . . . 8 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
54	53	anbi1i 624	. . . . . . 7 ⊢ ((𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷) ↔ (¬ 𝑥 = 𝑦 ∧ 𝐶 = 𝐷))
55		pm4.61 404	. . . . . . 7 ⊢ (¬ (𝐶 = 𝐷 → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷 ∧ ¬ 𝑥 = 𝑦))
56	52, 54, 55	3bitr4i 303	. . . . . 6 ⊢ ((𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷) ↔ ¬ (𝐶 = 𝐷 → 𝑥 = 𝑦))
57	56	rexbii 3081	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷) ↔ ∃𝑦 ∈ 𝐴 ¬ (𝐶 = 𝐷 → 𝑥 = 𝑦))
58		rexnal 3086	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝐶 = 𝐷 → 𝑥 = 𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦))
59	57, 58	bitri 275	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦))
60	59	rexbii 3081	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦))
61		rexnal 3086	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦))
62	60, 61	bitri 275	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦))
63	51, 62	sylibr 234	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  Vcvv 3438  ⦋csb 3847   class class class wbr 5096   ≼ cdom 8879   ≺ csdm 8880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884

This theorem is referenced by:  fphpdo  43001  pellex  43019