Theorem cpcolld 44227

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 3492	. . . . . . 7 ⊢ 𝑦 ∈ V
4		breq2 5170	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥𝐹𝑧 ↔ 𝑥𝐹𝑦))
5	3, 4	elab 3694	. . . . . 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 44215	. . 3 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧})
9		ssiun2 5070	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → Scott {𝑧 ∣ 𝑥𝐹𝑧} ⊆ ∪ 𝑥 ∈ 𝐴 Scott {𝑧 ∣ 𝑥𝐹𝑧})
10		dfcoll2 44221	. . . . . . . 8 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑧 ∣ 𝑥𝐹𝑧}
11	9, 10	sseqtrrdi 4060	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → Scott {𝑧 ∣ 𝑥𝐹𝑧} ⊆ (𝐹 Coll 𝐴))
12	11	sselda 4008	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ Scott {𝑧 ∣ 𝑥𝐹𝑧}) → 𝑦 ∈ (𝐹 Coll 𝐴))
13	4	elscottab 44213	. . . . . . 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 1916	. . 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 3077	. 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 1777   ∈ wcel 2108  {cab 2717  ∃wrex 3076  ∪ ciun 5015   class class class wbr 5166  Scott cscott 44204   Coll ccoll 44219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-r1 9833  df-rank 9834  df-scott 44205  df-coll 44220

This theorem is referenced by:  cpcoll2d  44228