Theorem cpcolld 44859

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 3467	. . . . . . 7 ⊢ 𝑦 ∈ V
4		breq2 5117	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥𝐹𝑧 ↔ 𝑥𝐹𝑦))
5	3, 4	elab 3647	. . . . . 6 ⊢ (𝑦 ∈ {𝑧 ∣ 𝑥𝐹𝑧} ↔ 𝑥𝐹𝑦)
6	2, 5	sylibr 237	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ {𝑧 ∣ 𝑥𝐹𝑧})
7	6	19.8ad 2224	. . . 4 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ {𝑧 ∣ 𝑥𝐹𝑧})
8	7	scotteld 9871	. . 3 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧})
9		ssiun2 5016	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → Scott {𝑧 ∣ 𝑥𝐹𝑧} ⊆ ∪ 𝑥 ∈ 𝐴 Scott {𝑧 ∣ 𝑥𝐹𝑧})
10		dfcoll2 44853	. . . . . . . 8 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑧 ∣ 𝑥𝐹𝑧}
11	9, 10	sseqtrrdi 3986	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → Scott {𝑧 ∣ 𝑥𝐹𝑧} ⊆ (𝐹 Coll 𝐴))
12	11	sselda 3945	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧}) → 𝑦 ∈ (𝐹 Coll 𝐴))
13	4	elscottab 9869	. . . . . . 7 ⊢ (𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧} → 𝑥𝐹𝑦)
14	13	adantl 486	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧}) → 𝑥𝐹𝑦)
15	12, 14	jca 520	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧}) → (𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦))
16	15	ex 417	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧} → (𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦)))
17	16	eximdv 1944	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧} → ∃𝑦(𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦)))
18	1, 8, 17	sylc 66	. 2 ⊢ (𝜑 → ∃𝑦(𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦))
19		df-rex 3096	. 2 ⊢ (∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦 ↔ ∃𝑦(𝑦 ∈ (𝐹 Coll 𝐴) ∧ 𝑥𝐹𝑦))
20	18, 19	sylibr 237	1 ⊢ (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∃wrex 3095  ∪ ciun 4960   class class class wbr 5113  Scott cscott 9856   Coll ccoll 44851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-r1 9735  df-rank 9736  df-scott 9857  df-coll 44852

This theorem is referenced by:  cpcoll2d  44860