Theorem prdsvallem 13474

Description: Lemma for prdsval 13475. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 13475, dependency on df-hom 13303 removed. (Revised by AV, 13-Oct-2024.)

Assertion

Ref	Expression
prdsvallem	⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V

Distinct variable groups:   𝑥,𝑟   𝑓,𝑔,𝑟   𝑣,𝑓,𝑔

Proof of Theorem prdsvallem

Step	Hyp	Ref	Expression
1		vex 2815	. 2 ⊢ 𝑣 ∈ V
2		fnmap 6888	. . . 4 ⊢ ↑𝑚 Fn (V × V)
3		vex 2815	. . . . . . . . . 10 ⊢ 𝑟 ∈ V
4	3	rnex 5024	. . . . . . . . 9 ⊢ ran 𝑟 ∈ V
5	4	uniex 4557	. . . . . . . 8 ⊢ ∪ ran 𝑟 ∈ V
6	5	rnex 5024	. . . . . . 7 ⊢ ran ∪ ran 𝑟 ∈ V
7	6	uniex 4557	. . . . . 6 ⊢ ∪ ran ∪ ran 𝑟 ∈ V
8	7	rnex 5024	. . . . 5 ⊢ ran ∪ ran ∪ ran 𝑟 ∈ V
9	8	uniex 4557	. . . 4 ⊢ ∪ ran ∪ ran ∪ ran 𝑟 ∈ V
10	3	dmex 5023	. . . 4 ⊢ dom 𝑟 ∈ V
11		fnovex 6082	. . . 4 ⊢ (( ↑𝑚 Fn (V × V) ∧ ∪ ran ∪ ran ∪ ran 𝑟 ∈ V ∧ dom 𝑟 ∈ V) → (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V)
12	2, 9, 10, 11	mp3an 1374	. . 3 ⊢ (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V
13	12	pwex 4295	. 2 ⊢ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V
14		vex 2815	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
15		vex 2815	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
16	14, 15	fvex 5689	. . . . . . . . 9 ⊢ (𝑓‘𝑥) ∈ V
17		vex 2815	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
18	17, 15	fvex 5689	. . . . . . . . 9 ⊢ (𝑔‘𝑥) ∈ V
19		ovssunirng 6084	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) ∈ V ∧ (𝑔‘𝑥) ∈ V) → ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (Hom ‘(𝑟‘𝑥)))
20	16, 18, 19	mp2an 426	. . . . . . . 8 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (Hom ‘(𝑟‘𝑥))
21		homid 13436	. . . . . . . . . . . 12 ⊢ Hom = Slot (Hom ‘ndx)
22	3, 15	fvex 5689	. . . . . . . . . . . . 13 ⊢ (𝑟‘𝑥) ∈ V
23	22	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑟‘𝑥) ∈ V)
24		homslid 13437	. . . . . . . . . . . . . 14 ⊢ (Hom = Slot (Hom ‘ndx) ∧ (Hom ‘ndx) ∈ ℕ)
25	24	simpri 113	. . . . . . . . . . . . 13 ⊢ (Hom ‘ndx) ∈ ℕ
26	25	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → (Hom ‘ndx) ∈ ℕ)
27	21, 23, 26	strfvssn 13223	. . . . . . . . . . 11 ⊢ (⊤ → (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥))
28	27	mptru 1407	. . . . . . . . . 10 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥)
29		fvssunirng 5684	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → (𝑟‘𝑥) ⊆ ∪ ran 𝑟)
30	29	elv 2816	. . . . . . . . . . 11 ⊢ (𝑟‘𝑥) ⊆ ∪ ran 𝑟
31		rnss 4986	. . . . . . . . . . 11 ⊢ ((𝑟‘𝑥) ⊆ ∪ ran 𝑟 → ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟)
32		uniss 3934	. . . . . . . . . . 11 ⊢ (ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟 → ∪ ran (𝑟‘𝑥) ⊆ ∪ ran ∪ ran 𝑟)
33	30, 31, 32	mp2b 8	. . . . . . . . . 10 ⊢ ∪ ran (𝑟‘𝑥) ⊆ ∪ ran ∪ ran 𝑟
34	28, 33	sstri 3246	. . . . . . . . 9 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟
35		rnss 4986	. . . . . . . . 9 ⊢ ((Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 → ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟)
36		uniss 3934	. . . . . . . . 9 ⊢ (ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟 → ∪ ran (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟)
37	34, 35, 36	mp2b 8	. . . . . . . 8 ⊢ ∪ ran (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
38	20, 37	sstri 3246	. . . . . . 7 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
39	38	rgenw 2597	. . . . . 6 ⊢ ∀𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
40		ss2ixp 6945	. . . . . 6 ⊢ (∀𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟 → X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟)
41	39, 40	ax-mp 5	. . . . 5 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟
42	10, 9	ixpconst 6942	. . . . 5 ⊢ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟 = (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
43	41, 42	sseqtri 3271	. . . 4 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
44	12, 43	elpwi2 4269	. . 3 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
45	44	rgen2w 2598	. 2 ⊢ ∀𝑓 ∈ 𝑣 ∀𝑔 ∈ 𝑣 X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
46	1, 1, 13, 45	mpoexw 6408	1 ⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V

Syntax hints:   = wceq 1398  ⊤wtru 1399   ∈ wcel 2203  ∀wral 2520  Vcvv 2812   ⊆ wss 3210  𝒫 cpw 3668  ∪ cuni 3913   × cxp 4746  dom cdm 4748  ran crn 4749   Fn wfn 5346  ‘cfv 5351  (class class class)co 6049   ∈ cmpo 6051   ↑_𝑚 cmap 6881  Xcixp 6932  ℕcn 9233  ndxcnx 13198  Slot cslot 13200  Hom chom 13290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-map 6883  df-ixp 6933  df-sub 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-hom 13303

This theorem is referenced by:  prdsval  13475