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Theorem cpcolld 44282
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 3483 . . . . . . 7 𝑦 ∈ V
4 breq2 5146 . . . . . . 7 (𝑧 = 𝑦 → (𝑥𝐹𝑧𝑥𝐹𝑦))
53, 4elab 3678 . . . . . 6 (𝑦 ∈ {𝑧𝑥𝐹𝑧} ↔ 𝑥𝐹𝑦)
62, 5sylibr 234 . . . . 5 (𝜑𝑦 ∈ {𝑧𝑥𝐹𝑧})
7619.8ad 2181 . . . 4 (𝜑 → ∃𝑦 𝑦 ∈ {𝑧𝑥𝐹𝑧})
87scotteld 44270 . . 3 (𝜑 → ∃𝑦 𝑦 ∈ Scott {𝑧𝑥𝐹𝑧})
9 ssiun2 5046 . . . . . . . 8 (𝑥𝐴 → Scott {𝑧𝑥𝐹𝑧} ⊆ 𝑥𝐴 Scott {𝑧𝑥𝐹𝑧})
10 dfcoll2 44276 . . . . . . . 8 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott {𝑧𝑥𝐹𝑧}
119, 10sseqtrrdi 4024 . . . . . . 7 (𝑥𝐴 → Scott {𝑧𝑥𝐹𝑧} ⊆ (𝐹 Coll 𝐴))
1211sselda 3982 . . . . . 6 ((𝑥𝐴𝑦 ∈ Scott {𝑧𝑥𝐹𝑧}) → 𝑦 ∈ (𝐹 Coll 𝐴))
134elscottab 44268 . . . . . . 7 (𝑦 ∈ Scott {𝑧𝑥𝐹𝑧} → 𝑥𝐹𝑦)
1413adantl 481 . . . . . 6 ((𝑥𝐴𝑦 ∈ Scott {𝑧𝑥𝐹𝑧}) → 𝑥𝐹𝑦)
1512, 14jca 511 . . . . 5 ((𝑥𝐴𝑦 ∈ Scott {𝑧𝑥𝐹𝑧}) → (𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦))
1615ex 412 . . . 4 (𝑥𝐴 → (𝑦 ∈ Scott {𝑧𝑥𝐹𝑧} → (𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦)))
1716eximdv 1916 . . 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 3070 . 2 (∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦 ↔ ∃𝑦(𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦))
2018, 19sylibr 234 1 (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1778  wcel 2107  {cab 2713  wrex 3069   ciun 4990   class class class wbr 5142  Scott cscott 44259   Coll ccoll 44274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-r1 9805  df-rank 9806  df-scott 44260  df-coll 44275
This theorem is referenced by:  cpcoll2d  44283
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