Theorem cpcolld 44249

Description: Property of the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)

Hypotheses

Ref	Expression
cpcolld.1	⊢ (𝜑 → 𝑥 ∈ 𝐴)
cpcolld.2	⊢ (𝜑 → 𝑥𝐹𝑦)

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.

Step	Hyp	Ref	Expression
1		cpcolld.1	. . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
2		cpcolld.2	. . . . . 6 ⊢ (𝜑 → 𝑥𝐹𝑦)
3		vex 3468	. . . . . . 7 ⊢ 𝑦 ∈ V
4		breq2 5128	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥𝐹𝑧 ↔ 𝑥𝐹𝑦))
5	3, 4	elab 3663	. . . . . 6 ⊢ (𝑦 ∈ {𝑧 ∣ 𝑥𝐹𝑧} ↔ 𝑥𝐹𝑦)
6	2, 5	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ {𝑧 ∣ 𝑥𝐹𝑧})
7	6	19.8ad 2183	. . . 4 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ {𝑧 ∣ 𝑥𝐹𝑧})
8	7	scotteld 44237	. . 3 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧})
9		ssiun2 5028	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → Scott {𝑧 ∣ 𝑥𝐹𝑧} ⊆ ∪ 𝑥 ∈ 𝐴 Scott {𝑧 ∣ 𝑥𝐹𝑧})
10		dfcoll2 44243	. . . . . . . 8 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑧 ∣ 𝑥𝐹𝑧}
11	9, 10	sseqtrrdi 4005	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → Scott {𝑧 ∣ 𝑥𝐹𝑧} ⊆ (𝐹 Coll 𝐴))
12	11	sselda 3963	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧}) → 𝑦 ∈ (𝐹 Coll 𝐴))
13	4	elscottab 44235	. . . . . . 7 ⊢ (𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧} → 𝑥𝐹𝑦)
14	13	adantl 481	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧}) → 𝑥𝐹𝑦)
15	12, 14	jca 511	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧}) → (𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦))
16	15	ex 412	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧} → (𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦)))
17	16	eximdv 1917	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧} → ∃𝑦(𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦)))
18	1, 8, 17	sylc 65	. 2 ⊢ (𝜑 → ∃𝑦(𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦))
19		df-rex 3062	. 2 ⊢ (∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦 ↔ ∃𝑦(𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦))
20	18, 19	sylibr 234	1 ⊢ (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  {cab 2714  ∃wrex 3061  ∪ ciun 4972   class class class wbr 5124  Scott cscott 44226   Coll ccoll 44241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-r1 9783  df-rank 9784  df-scott 44227  df-coll 44242

This theorem is referenced by:  cpcoll2d  44250