Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fphpd Structured version   Visualization version   GIF version

Theorem fphpd 41125
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 (𝜑𝐵𝐴)
fphpd.b ((𝜑𝑥𝐴) → 𝐶𝐵)
fphpd.c (𝑥 = 𝑦𝐶 = 𝐷)
Assertion
Ref Expression
fphpd (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝐶 = 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.
StepHypRef Expression
1 domnsym 9043 . . . 4 (𝐴𝐵 → ¬ 𝐵𝐴)
2 fphpd.a . . . 4 (𝜑𝐵𝐴)
31, 2nsyl3 138 . . 3 (𝜑 → ¬ 𝐴𝐵)
4 relsdom 8890 . . . . . . 7 Rel ≺
54brrelex1i 5688 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
62, 5syl 17 . . . . 5 (𝜑𝐵 ∈ V)
76adantr 481 . . . 4 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → 𝐵 ∈ V)
8 nfv 1917 . . . . . . . . 9 𝑥(𝜑𝑎𝐴)
9 nfcsb1v 3880 . . . . . . . . . 10 𝑥𝑎 / 𝑥𝐶
109nfel1 2923 . . . . . . . . 9 𝑥𝑎 / 𝑥𝐶𝐵
118, 10nfim 1899 . . . . . . . 8 𝑥((𝜑𝑎𝐴) → 𝑎 / 𝑥𝐶𝐵)
12 eleq1w 2820 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
1312anbi2d 629 . . . . . . . . 9 (𝑥 = 𝑎 → ((𝜑𝑥𝐴) ↔ (𝜑𝑎𝐴)))
14 csbeq1a 3869 . . . . . . . . . 10 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
1514eleq1d 2822 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐶𝐵𝑎 / 𝑥𝐶𝐵))
1613, 15imbi12d 344 . . . . . . . 8 (𝑥 = 𝑎 → (((𝜑𝑥𝐴) → 𝐶𝐵) ↔ ((𝜑𝑎𝐴) → 𝑎 / 𝑥𝐶𝐵)))
17 fphpd.b . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐶𝐵)
1811, 16, 17chvarfv 2233 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎 / 𝑥𝐶𝐵)
1918ex 413 . . . . . 6 (𝜑 → (𝑎𝐴𝑎 / 𝑥𝐶𝐵))
2019adantr 481 . . . . 5 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → (𝑎𝐴𝑎 / 𝑥𝐶𝐵))
21 csbid 3868 . . . . . . . . . . 11 𝑥 / 𝑥𝐶 = 𝐶
22 vex 3449 . . . . . . . . . . . 12 𝑦 ∈ V
23 fphpd.c . . . . . . . . . . . 12 (𝑥 = 𝑦𝐶 = 𝐷)
2422, 23csbie 3891 . . . . . . . . . . 11 𝑦 / 𝑥𝐶 = 𝐷
2521, 24eqeq12i 2754 . . . . . . . . . 10 (𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝐶 = 𝐷)
2625imbi1i 349 . . . . . . . . 9 ((𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦))
27262ralbii 3127 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴 (𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
28 nfcsb1v 3880 . . . . . . . . . . . 12 𝑥𝑦 / 𝑥𝐶
299, 28nfeq 2920 . . . . . . . . . . 11 𝑥𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶
30 nfv 1917 . . . . . . . . . . 11 𝑥 𝑎 = 𝑦
3129, 30nfim 1899 . . . . . . . . . 10 𝑥(𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 = 𝑦)
32 nfv 1917 . . . . . . . . . 10 𝑦(𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)
33 csbeq1 3858 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 / 𝑥𝐶 = 𝑎 / 𝑥𝐶)
3433eqeq1d 2738 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶))
35 equequ1 2028 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑥 = 𝑦𝑎 = 𝑦))
3634, 35imbi12d 344 . . . . . . . . . 10 (𝑥 = 𝑎 → ((𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑥 = 𝑦) ↔ (𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 = 𝑦)))
37 csbeq1 3858 . . . . . . . . . . . 12 (𝑦 = 𝑏𝑦 / 𝑥𝐶 = 𝑏 / 𝑥𝐶)
3837eqeq2d 2747 . . . . . . . . . . 11 (𝑦 = 𝑏 → (𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶))
39 equequ2 2029 . . . . . . . . . . 11 (𝑦 = 𝑏 → (𝑎 = 𝑦𝑎 = 𝑏))
4038, 39imbi12d 344 . . . . . . . . . 10 (𝑦 = 𝑏 → ((𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 = 𝑦) ↔ (𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)))
4131, 32, 36, 40rspc2 3588 . . . . . . . . 9 ((𝑎𝐴𝑏𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑥 = 𝑦) → (𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)))
4241com12 32 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴 (𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑥 = 𝑦) → ((𝑎𝐴𝑏𝐴) → (𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)))
4327, 42sylbir 234 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦) → ((𝑎𝐴𝑏𝐴) → (𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)))
44 id 22 . . . . . . . 8 ((𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏) → (𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏))
45 csbeq1 3858 . . . . . . . 8 (𝑎 = 𝑏𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶)
4644, 45impbid1 224 . . . . . . 7 ((𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏) → (𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏))
4743, 46syl6 35 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦) → ((𝑎𝐴𝑏𝐴) → (𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)))
4847adantl 482 . . . . 5 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → ((𝑎𝐴𝑏𝐴) → (𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)))
4920, 48dom2d 8933 . . . 4 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → (𝐵 ∈ V → 𝐴𝐵))
507, 49mpd 15 . . 3 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → 𝐴𝐵)
513, 50mtand 814 . 2 (𝜑 → ¬ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
52 ancom 461 . . . . . . 7 ((¬ 𝑥 = 𝑦𝐶 = 𝐷) ↔ (𝐶 = 𝐷 ∧ ¬ 𝑥 = 𝑦))
53 df-ne 2944 . . . . . . . 8 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
5453anbi1i 624 . . . . . . 7 ((𝑥𝑦𝐶 = 𝐷) ↔ (¬ 𝑥 = 𝑦𝐶 = 𝐷))
55 pm4.61 405 . . . . . . 7 (¬ (𝐶 = 𝐷𝑥 = 𝑦) ↔ (𝐶 = 𝐷 ∧ ¬ 𝑥 = 𝑦))
5652, 54, 553bitr4i 302 . . . . . 6 ((𝑥𝑦𝐶 = 𝐷) ↔ ¬ (𝐶 = 𝐷𝑥 = 𝑦))
5756rexbii 3097 . . . . 5 (∃𝑦𝐴 (𝑥𝑦𝐶 = 𝐷) ↔ ∃𝑦𝐴 ¬ (𝐶 = 𝐷𝑥 = 𝑦))
58 rexnal 3103 . . . . 5 (∃𝑦𝐴 ¬ (𝐶 = 𝐷𝑥 = 𝑦) ↔ ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
5957, 58bitri 274 . . . 4 (∃𝑦𝐴 (𝑥𝑦𝐶 = 𝐷) ↔ ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
6059rexbii 3097 . . 3 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝐶 = 𝐷) ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
61 rexnal 3103 . . 3 (∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
6260, 61bitri 274 . 2 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝐶 = 𝐷) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
6351, 62sylibr 233 1 (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  csb 3855   class class class wbr 5105  cdom 8881  csdm 8882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886
This theorem is referenced by:  fphpdo  41126  pellex  41144
  Copyright terms: Public domain W3C validator