Theorem cpcolld 44247

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 3451	. . . . . . 7 ⊢ 𝑦 ∈ V
4		breq2 5111	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥𝐹𝑧 ↔ 𝑥𝐹𝑦))
5	3, 4	elab 3646	. . . . . 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 44235	. . 3 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧})
9		ssiun2 5011	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → Scott {𝑧 ∣ 𝑥𝐹𝑧} ⊆ ∪ 𝑥 ∈ 𝐴 Scott {𝑧 ∣ 𝑥𝐹𝑧})
10		dfcoll2 44241	. . . . . . . 8 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑧 ∣ 𝑥𝐹𝑧}
11	9, 10	sseqtrrdi 3988	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → Scott {𝑧 ∣ 𝑥𝐹𝑧} ⊆ (𝐹 Coll 𝐴))
12	11	sselda 3946	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧}) → 𝑦 ∈ (𝐹 Coll 𝐴))
13	4	elscottab 44233	. . . . . . 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 3054	. 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 2707  ∃wrex 3053  ∪ ciun 4955   class class class wbr 5107  Scott cscott 44224   Coll ccoll 44239

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 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-r1 9717  df-rank 9718  df-scott 44225  df-coll 44240

This theorem is referenced by:  cpcoll2d  44248