Theorem grucollcld 40671

Description: A Grothendieck universe contains the output of a collection operation whenever its left input is a relation on the universe, and its right input is in the universe. (Contributed by Rohan Ridenour, 11-Aug-2023.)

Hypotheses

Ref	Expression
grucollcld.1	⊢ (𝜑 → 𝐺 ∈ Univ)
grucollcld.2	⊢ (𝜑 → 𝐹 ⊆ (𝐺 × 𝐺))
grucollcld.3	⊢ (𝜑 → 𝐴 ∈ 𝐺)

Assertion

Ref	Expression
grucollcld	⊢ (𝜑 → (𝐹 Coll 𝐴) ∈ 𝐺)

Proof of Theorem grucollcld

Dummy variables 𝑥 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcoll2 40663	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦}
2		grucollcld.1	. . 3 ⊢ (𝜑 → 𝐺 ∈ Univ)
3		grucollcld.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐺)
4		simpr 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅)
5	2	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → 𝐺 ∈ Univ)
6	3	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → 𝐴 ∈ 𝐺)
7	5, 6	gru0eld 40640	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → ∅ ∈ 𝐺)
8	4, 7	eqeltrd 2912	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺)
9		neq0 4302	. . . . . . 7 ⊢ (¬ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅ ↔ ∃𝑧 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦})
10	2	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝐺 ∈ Univ)
11		breq2 5063	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑦 ↔ 𝑥𝐹𝑧))
12	11	elscottab 40655	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦} → 𝑥𝐹𝑧)
13	12	adantl 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝑥𝐹𝑧)
14		grucollcld.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ⊆ (𝐺 × 𝐺))
15	14	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝐹 ⊆ (𝐺 × 𝐺))
16	15	ssbrd 5102	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → (𝑥𝐹𝑧 → 𝑥(𝐺 × 𝐺)𝑧))
17	13, 16	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝑥(𝐺 × 𝐺)𝑧)
18		brxp 5594	. . . . . . . . . . . 12 ⊢ (𝑥(𝐺 × 𝐺)𝑧 ↔ (𝑥 ∈ 𝐺 ∧ 𝑧 ∈ 𝐺))
19	18	simprbi 499	. . . . . . . . . . 11 ⊢ (𝑥(𝐺 × 𝐺)𝑧 → 𝑧 ∈ 𝐺)
20	17, 19	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝑧 ∈ 𝐺)
21		simpr 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦})
22	10, 20, 21	gruscottcld 40660	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺)
23	22	expcom 416	. . . . . . . 8 ⊢ (𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦} → ((𝜑 ∧ 𝑥 ∈ 𝐴) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺))
24	23	exlimiv 1930	. . . . . . 7 ⊢ (∃𝑧 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦} → ((𝜑 ∧ 𝑥 ∈ 𝐴) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺))
25	9, 24	sylbi 219	. . . . . 6 ⊢ (¬ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅ → ((𝜑 ∧ 𝑥 ∈ 𝐴) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺))
26	25	impcom 410	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺)
27	8, 26	pm2.61dan 811	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺)
28	27	ralrimiva 3181	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺)
29		gruiun 10214	. . 3 ⊢ ((𝐺 ∈ Univ ∧ 𝐴 ∈ 𝐺 ∧ ∀𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺) → ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺)
30	2, 3, 28, 29	syl3anc 1366	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺)
31	1, 30	eqeltrid 2916	1 ⊢ (𝜑 → (𝐹 Coll 𝐴) ∈ 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1536  ∃wex 1779   ∈ wcel 2113  {cab 2798  ∀wral 3137   ⊆ wss 3929  ∅c0 4284  ∪ ciun 4912   class class class wbr 5059   × cxp 5546  Univcgru 10205  Scott cscott 40646   Coll ccoll 40661

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-reg 9049  ax-inf2 9097  ax-ac2 9878

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-1o 8095  df-er 8282  df-map 8401  df-en 8503  df-dom 8504  df-sdom 8505  df-fin 8506  df-tc 9172  df-r1 9186  df-rank 9187  df-card 9361  df-cf 9363  df-acn 9364  df-ac 9535  df-wina 10099  df-ina 10100  df-gru 10206  df-scott 40647  df-coll 40662

This theorem is referenced by:  grumnudlem  40696