Users' Mathboxes Mathbox for Rohan Ridenour < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cpcolld Structured version   Visualization version   GIF version

Theorem cpcolld 42630
Description: Property of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypotheses
Ref Expression
cpcolld.1 (𝜑𝑥𝐴)
cpcolld.2 (𝜑𝑥𝐹𝑦)
Assertion
Ref Expression
cpcolld (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem cpcolld
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cpcolld.1 . . 3 (𝜑𝑥𝐴)
2 cpcolld.2 . . . . . 6 (𝜑𝑥𝐹𝑦)
3 vex 3451 . . . . . . 7 𝑦 ∈ V
4 breq2 5113 . . . . . . 7 (𝑧 = 𝑦 → (𝑥𝐹𝑧𝑥𝐹𝑦))
53, 4elab 3634 . . . . . 6 (𝑦 ∈ {𝑧𝑥𝐹𝑧} ↔ 𝑥𝐹𝑦)
62, 5sylibr 233 . . . . 5 (𝜑𝑦 ∈ {𝑧𝑥𝐹𝑧})
7619.8ad 2176 . . . 4 (𝜑 → ∃𝑦 𝑦 ∈ {𝑧𝑥𝐹𝑧})
87scotteld 42618 . . 3 (𝜑 → ∃𝑦 𝑦 ∈ Scott {𝑧𝑥𝐹𝑧})
9 ssiun2 5011 . . . . . . . 8 (𝑥𝐴 → Scott {𝑧𝑥𝐹𝑧} ⊆ 𝑥𝐴 Scott {𝑧𝑥𝐹𝑧})
10 dfcoll2 42624 . . . . . . . 8 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott {𝑧𝑥𝐹𝑧}
119, 10sseqtrrdi 3999 . . . . . . 7 (𝑥𝐴 → Scott {𝑧𝑥𝐹𝑧} ⊆ (𝐹 Coll 𝐴))
1211sselda 3948 . . . . . 6 ((𝑥𝐴𝑦 ∈ Scott {𝑧𝑥𝐹𝑧}) → 𝑦 ∈ (𝐹 Coll 𝐴))
134elscottab 42616 . . . . . . 7 (𝑦 ∈ Scott {𝑧𝑥𝐹𝑧} → 𝑥𝐹𝑦)
1413adantl 483 . . . . . 6 ((𝑥𝐴𝑦 ∈ Scott {𝑧𝑥𝐹𝑧}) → 𝑥𝐹𝑦)
1512, 14jca 513 . . . . 5 ((𝑥𝐴𝑦 ∈ Scott {𝑧𝑥𝐹𝑧}) → (𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦))
1615ex 414 . . . 4 (𝑥𝐴 → (𝑦 ∈ Scott {𝑧𝑥𝐹𝑧} → (𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦)))
1716eximdv 1921 . . 3 (𝑥𝐴 → (∃𝑦 𝑦 ∈ Scott {𝑧𝑥𝐹𝑧} → ∃𝑦(𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦)))
181, 8, 17sylc 65 . 2 (𝜑 → ∃𝑦(𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦))
19 df-rex 3071 . 2 (∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦 ↔ ∃𝑦(𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦))
2018, 19sylibr 233 1 (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wex 1782  wcel 2107  {cab 2710  wrex 3070   ciun 4958   class class class wbr 5109  Scott cscott 42607   Coll ccoll 42622
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-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-r1 9708  df-rank 9709  df-scott 42608  df-coll 42623
This theorem is referenced by:  cpcoll2d  42631
  Copyright terms: Public domain W3C validator