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Theorem fphpd 42709
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 9161 . . . 4 (𝐴𝐵 → ¬ 𝐵𝐴)
2 fphpd.a . . . 4 (𝜑𝐵𝐴)
31, 2nsyl3 138 . . 3 (𝜑 → ¬ 𝐴𝐵)
4 relsdom 9006 . . . . . . 7 Rel ≺
54brrelex1i 5755 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
62, 5syl 17 . . . . 5 (𝜑𝐵 ∈ V)
76adantr 480 . . . 4 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → 𝐵 ∈ V)
8 nfv 1913 . . . . . . . . 9 𝑥(𝜑𝑎𝐴)
9 nfcsb1v 3940 . . . . . . . . . 10 𝑥𝑎 / 𝑥𝐶
109nfel1 2921 . . . . . . . . 9 𝑥𝑎 / 𝑥𝐶𝐵
118, 10nfim 1895 . . . . . . . 8 𝑥((𝜑𝑎𝐴) → 𝑎 / 𝑥𝐶𝐵)
12 eleq1w 2821 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
1312anbi2d 629 . . . . . . . . 9 (𝑥 = 𝑎 → ((𝜑𝑥𝐴) ↔ (𝜑𝑎𝐴)))
14 csbeq1a 3929 . . . . . . . . . 10 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
1514eleq1d 2823 . . . . . . . . 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 2236 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎 / 𝑥𝐶𝐵)
1918ex 412 . . . . . 6 (𝜑 → (𝑎𝐴𝑎 / 𝑥𝐶𝐵))
2019adantr 480 . . . . 5 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → (𝑎𝐴𝑎 / 𝑥𝐶𝐵))
21 csbid 3928 . . . . . . . . . . 11 𝑥 / 𝑥𝐶 = 𝐶
22 vex 3486 . . . . . . . . . . . 12 𝑦 ∈ V
23 fphpd.c . . . . . . . . . . . 12 (𝑥 = 𝑦𝐶 = 𝐷)
2422, 23csbie 3951 . . . . . . . . . . 11 𝑦 / 𝑥𝐶 = 𝐷
2521, 24eqeq12i 2752 . . . . . . . . . 10 (𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝐶 = 𝐷)
2625imbi1i 349 . . . . . . . . 9 ((𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦))
27262ralbii 3130 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴 (𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
28 nfcsb1v 3940 . . . . . . . . . . . 12 𝑥𝑦 / 𝑥𝐶
299, 28nfeq 2918 . . . . . . . . . . 11 𝑥𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶
30 nfv 1913 . . . . . . . . . . 11 𝑥 𝑎 = 𝑦
3129, 30nfim 1895 . . . . . . . . . 10 𝑥(𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 = 𝑦)
32 nfv 1913 . . . . . . . . . 10 𝑦(𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)
33 csbeq1 3918 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 / 𝑥𝐶 = 𝑎 / 𝑥𝐶)
3433eqeq1d 2736 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶))
35 equequ1 2024 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑥 = 𝑦𝑎 = 𝑦))
3634, 35imbi12d 344 . . . . . . . . . 10 (𝑥 = 𝑎 → ((𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑥 = 𝑦) ↔ (𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 = 𝑦)))
37 csbeq1 3918 . . . . . . . . . . . 12 (𝑦 = 𝑏𝑦 / 𝑥𝐶 = 𝑏 / 𝑥𝐶)
3837eqeq2d 2745 . . . . . . . . . . 11 (𝑦 = 𝑏 → (𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶))
39 equequ2 2025 . . . . . . . . . . 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 3640 . . . . . . . . 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 235 . . . . . . 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 3918 . . . . . . . 8 (𝑎 = 𝑏𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶)
4644, 45impbid1 225 . . . . . . 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 481 . . . . 5 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → ((𝑎𝐴𝑏𝐴) → (𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)))
4920, 48dom2d 9049 . . . 4 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → (𝐵 ∈ V → 𝐴𝐵))
507, 49mpd 15 . . 3 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → 𝐴𝐵)
513, 50mtand 815 . 2 (𝜑 → ¬ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
52 ancom 460 . . . . . . 7 ((¬ 𝑥 = 𝑦𝐶 = 𝐷) ↔ (𝐶 = 𝐷 ∧ ¬ 𝑥 = 𝑦))
53 df-ne 2943 . . . . . . . 8 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
5453anbi1i 623 . . . . . . 7 ((𝑥𝑦𝐶 = 𝐷) ↔ (¬ 𝑥 = 𝑦𝐶 = 𝐷))
55 pm4.61 404 . . . . . . 7 (¬ (𝐶 = 𝐷𝑥 = 𝑦) ↔ (𝐶 = 𝐷 ∧ ¬ 𝑥 = 𝑦))
5652, 54, 553bitr4i 303 . . . . . 6 ((𝑥𝑦𝐶 = 𝐷) ↔ ¬ (𝐶 = 𝐷𝑥 = 𝑦))
5756rexbii 3096 . . . . 5 (∃𝑦𝐴 (𝑥𝑦𝐶 = 𝐷) ↔ ∃𝑦𝐴 ¬ (𝐶 = 𝐷𝑥 = 𝑦))
58 rexnal 3102 . . . . 5 (∃𝑦𝐴 ¬ (𝐶 = 𝐷𝑥 = 𝑦) ↔ ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
5957, 58bitri 275 . . . 4 (∃𝑦𝐴 (𝑥𝑦𝐶 = 𝐷) ↔ ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
6059rexbii 3096 . . 3 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝐶 = 𝐷) ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
61 rexnal 3102 . . 3 (∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
6260, 61bitri 275 . 2 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝐶 = 𝐷) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
6351, 62sylibr 234 1 (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2103  wne 2942  wral 3063  wrex 3072  Vcvv 3482  csb 3915   class class class wbr 5169  cdom 8997  csdm 8998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-er 8759  df-en 9000  df-dom 9001  df-sdom 9002
This theorem is referenced by:  fphpdo  42710  pellex  42728
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