Theorem fphpd 42158

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 9115	. . . 4 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴)
2		fphpd.a	. . . 4 ⊢ (𝜑 → 𝐵 ≺ 𝐴)
3	1, 2	nsyl3 138	. . 3 ⊢ (𝜑 → ¬ 𝐴 ≼ 𝐵)
4		relsdom 8962	. . . . . . 7 ⊢ Rel ≺
5	4	brrelex1i 5728	. . . . . 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 1910	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ 𝐴)
9		nfcsb1v 3914	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
10	9	nfel1 2914	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 ∈ 𝐵
11	8, 10	nfim 1892	. . . . . . . 8 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 ∈ 𝐵)
12		eleq1w 2811	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
13	12	anbi2d 628	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑎 ∈ 𝐴)))
14		csbeq1a 3903	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
15	14	eleq1d 2813	. . . . . . . . 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 2226	. . . . . . 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 3902	. . . . . . . . . . 11 ⊢ ⦋𝑥 / 𝑥⦌𝐶 = 𝐶
22		vex 3473	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
23		fphpd.c	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
24	22, 23	csbie 3925	. . . . . . . . . . 11 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐷
25	21, 24	eqeq12i 2745	. . . . . . . . . 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 3123	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (⦋𝑥 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦))
28		nfcsb1v 3914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
29	9, 28	nfeq 2911	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶
30		nfv 1910	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑎 = 𝑦
31	29, 30	nfim 1892	. . . . . . . . . 10 ⊢ Ⅎ𝑥(⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 → 𝑎 = 𝑦)
32		nfv 1910	. . . . . . . . . 10 ⊢ Ⅎ𝑦(⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶 → 𝑎 = 𝑏)
33		csbeq1 3892	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ⦋𝑥 / 𝑥⦌𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
34	33	eqeq1d 2729	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (⦋𝑥 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶))
35		equequ1 2021	. . . . . . . . . . 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 3892	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶)
38	37	eqeq2d 2738	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶))
39		equequ2 2022	. . . . . . . . . . 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 3616	. . . . . . . . 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 234	. . . . . . 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 3892	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑥⦌𝐶)
46	44, 45	impbid1 224	. . . . . . 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 9005	. . . 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 2936	. . . . . . . 8 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
54	53	anbi1i 623	. . . . . . 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 3089	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷) ↔ ∃𝑦 ∈ 𝐴 ¬ (𝐶 = 𝐷 → 𝑥 = 𝑦))
58		rexnal 3095	. . . . 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 3089	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦))
61		rexnal 3095	. . 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 233	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  ∀wral 3056  ∃wrex 3065  Vcvv 3469  ⦋csb 3889   class class class wbr 5142   ≼ cdom 8953   ≺ csdm 8954

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-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

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-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  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-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958

This theorem is referenced by:  fphpdo  42159  pellex  42177