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Theorem cpcolld 44254
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 3482 . . . . . . 7 𝑦 ∈ V
4 breq2 5152 . . . . . . 7 (𝑧 = 𝑦 → (𝑥𝐹𝑧𝑥𝐹𝑦))
53, 4elab 3681 . . . . . 6 (𝑦 ∈ {𝑧𝑥𝐹𝑧} ↔ 𝑥𝐹𝑦)
62, 5sylibr 234 . . . . 5 (𝜑𝑦 ∈ {𝑧𝑥𝐹𝑧})
7619.8ad 2180 . . . 4 (𝜑 → ∃𝑦 𝑦 ∈ {𝑧𝑥𝐹𝑧})
87scotteld 44242 . . 3 (𝜑 → ∃𝑦 𝑦 ∈ Scott {𝑧𝑥𝐹𝑧})
9 ssiun2 5052 . . . . . . . 8 (𝑥𝐴 → Scott {𝑧𝑥𝐹𝑧} ⊆ 𝑥𝐴 Scott {𝑧𝑥𝐹𝑧})
10 dfcoll2 44248 . . . . . . . 8 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott {𝑧𝑥𝐹𝑧}
119, 10sseqtrrdi 4047 . . . . . . 7 (𝑥𝐴 → Scott {𝑧𝑥𝐹𝑧} ⊆ (𝐹 Coll 𝐴))
1211sselda 3995 . . . . . 6 ((𝑥𝐴𝑦 ∈ Scott {𝑧𝑥𝐹𝑧}) → 𝑦 ∈ (𝐹 Coll 𝐴))
134elscottab 44240 . . . . . . 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 1915 . . 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 3069 . 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 1776  wcel 2106  {cab 2712  wrex 3068   ciun 4996   class class class wbr 5148  Scott cscott 44231   Coll ccoll 44246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803  df-scott 44232  df-coll 44247
This theorem is referenced by:  cpcoll2d  44255
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