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Theorem fphpd 41554
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 9099 . . . 4 (𝐴𝐵 → ¬ 𝐵𝐴)
2 fphpd.a . . . 4 (𝜑𝐵𝐴)
31, 2nsyl3 138 . . 3 (𝜑 → ¬ 𝐴𝐵)
4 relsdom 8946 . . . . . . 7 Rel ≺
54brrelex1i 5733 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
62, 5syl 17 . . . . 5 (𝜑𝐵 ∈ V)
76adantr 482 . . . 4 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → 𝐵 ∈ V)
8 nfv 1918 . . . . . . . . 9 𝑥(𝜑𝑎𝐴)
9 nfcsb1v 3919 . . . . . . . . . 10 𝑥𝑎 / 𝑥𝐶
109nfel1 2920 . . . . . . . . 9 𝑥𝑎 / 𝑥𝐶𝐵
118, 10nfim 1900 . . . . . . . 8 𝑥((𝜑𝑎𝐴) → 𝑎 / 𝑥𝐶𝐵)
12 eleq1w 2817 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
1312anbi2d 630 . . . . . . . . 9 (𝑥 = 𝑎 → ((𝜑𝑥𝐴) ↔ (𝜑𝑎𝐴)))
14 csbeq1a 3908 . . . . . . . . . 10 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
1514eleq1d 2819 . . . . . . . . 9 (𝑥 = 𝑎 → (𝐶𝐵𝑎 / 𝑥𝐶𝐵))
1613, 15imbi12d 345 . . . . . . . 8 (𝑥 = 𝑎 → (((𝜑𝑥𝐴) → 𝐶𝐵) ↔ ((𝜑𝑎𝐴) → 𝑎 / 𝑥𝐶𝐵)))
17 fphpd.b . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐶𝐵)
1811, 16, 17chvarfv 2234 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎 / 𝑥𝐶𝐵)
1918ex 414 . . . . . 6 (𝜑 → (𝑎𝐴𝑎 / 𝑥𝐶𝐵))
2019adantr 482 . . . . 5 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → (𝑎𝐴𝑎 / 𝑥𝐶𝐵))
21 csbid 3907 . . . . . . . . . . 11 𝑥 / 𝑥𝐶 = 𝐶
22 vex 3479 . . . . . . . . . . . 12 𝑦 ∈ V
23 fphpd.c . . . . . . . . . . . 12 (𝑥 = 𝑦𝐶 = 𝐷)
2422, 23csbie 3930 . . . . . . . . . . 11 𝑦 / 𝑥𝐶 = 𝐷
2521, 24eqeq12i 2751 . . . . . . . . . 10 (𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝐶 = 𝐷)
2625imbi1i 350 . . . . . . . . 9 ((𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦))
27262ralbii 3129 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴 (𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
28 nfcsb1v 3919 . . . . . . . . . . . 12 𝑥𝑦 / 𝑥𝐶
299, 28nfeq 2917 . . . . . . . . . . 11 𝑥𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶
30 nfv 1918 . . . . . . . . . . 11 𝑥 𝑎 = 𝑦
3129, 30nfim 1900 . . . . . . . . . 10 𝑥(𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 = 𝑦)
32 nfv 1918 . . . . . . . . . 10 𝑦(𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)
33 csbeq1 3897 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 / 𝑥𝐶 = 𝑎 / 𝑥𝐶)
3433eqeq1d 2735 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶))
35 equequ1 2029 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑥 = 𝑦𝑎 = 𝑦))
3634, 35imbi12d 345 . . . . . . . . . 10 (𝑥 = 𝑎 → ((𝑥 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑥 = 𝑦) ↔ (𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 = 𝑦)))
37 csbeq1 3897 . . . . . . . . . . . 12 (𝑦 = 𝑏𝑦 / 𝑥𝐶 = 𝑏 / 𝑥𝐶)
3837eqeq2d 2744 . . . . . . . . . . 11 (𝑦 = 𝑏 → (𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶))
39 equequ2 2030 . . . . . . . . . . 11 (𝑦 = 𝑏 → (𝑎 = 𝑦𝑎 = 𝑏))
4038, 39imbi12d 345 . . . . . . . . . 10 (𝑦 = 𝑏 → ((𝑎 / 𝑥𝐶 = 𝑦 / 𝑥𝐶𝑎 = 𝑦) ↔ (𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)))
4131, 32, 36, 40rspc2 3621 . . . . . . . . 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 3897 . . . . . . . 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 483 . . . . 5 ((𝜑 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → ((𝑎𝐴𝑏𝐴) → (𝑎 / 𝑥𝐶 = 𝑏 / 𝑥𝐶𝑎 = 𝑏)))
4920, 48dom2d 8989 . . . 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 462 . . . . . . 7 ((¬ 𝑥 = 𝑦𝐶 = 𝐷) ↔ (𝐶 = 𝐷 ∧ ¬ 𝑥 = 𝑦))
53 df-ne 2942 . . . . . . . 8 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
5453anbi1i 625 . . . . . . 7 ((𝑥𝑦𝐶 = 𝐷) ↔ (¬ 𝑥 = 𝑦𝐶 = 𝐷))
55 pm4.61 406 . . . . . . 7 (¬ (𝐶 = 𝐷𝑥 = 𝑦) ↔ (𝐶 = 𝐷 ∧ ¬ 𝑥 = 𝑦))
5652, 54, 553bitr4i 303 . . . . . 6 ((𝑥𝑦𝐶 = 𝐷) ↔ ¬ (𝐶 = 𝐷𝑥 = 𝑦))
5756rexbii 3095 . . . . 5 (∃𝑦𝐴 (𝑥𝑦𝐶 = 𝐷) ↔ ∃𝑦𝐴 ¬ (𝐶 = 𝐷𝑥 = 𝑦))
58 rexnal 3101 . . . . 5 (∃𝑦𝐴 ¬ (𝐶 = 𝐷𝑥 = 𝑦) ↔ ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
5957, 58bitri 275 . . . 4 (∃𝑦𝐴 (𝑥𝑦𝐶 = 𝐷) ↔ ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
6059rexbii 3095 . . 3 (∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝐶 = 𝐷) ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
61 rexnal 3101 . . 3 (∃𝑥𝐴 ¬ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦) ↔ ¬ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))
6260, 61bitri 275 . 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 397   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  Vcvv 3475  csb 3894   class class class wbr 5149  cdom 8937  csdm 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942
This theorem is referenced by:  fphpdo  41555  pellex  41573
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