Theorem prdsvallem 13623

Description: Lemma for prdsval 14175. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 14175, dependency on df-hom 13457 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 2824	. 2 ⊢ 𝑣 ∈ V
2		fnmap 6929	. . . 4 ⊢ ↑𝑚 Fn (V × V)
3		vex 2824	. . . . . . . . . 10 ⊢ 𝑟 ∈ V
4	3	rnex 5050	. . . . . . . . 9 ⊢ ran 𝑟 ∈ V
5	4	uniex 4583	. . . . . . . 8 ⊢ ∪ ran 𝑟 ∈ V
6	5	rnex 5050	. . . . . . 7 ⊢ ran ∪ ran 𝑟 ∈ V
7	6	uniex 4583	. . . . . 6 ⊢ ∪ ran ∪ ran 𝑟 ∈ V
8	7	rnex 5050	. . . . 5 ⊢ ran ∪ ran ∪ ran 𝑟 ∈ V
9	8	uniex 4583	. . . 4 ⊢ ∪ ran ∪ ran ∪ ran 𝑟 ∈ V
10	3	dmex 5049	. . . 4 ⊢ dom 𝑟 ∈ V
11		fnovex 6118	. . . 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 1378	. . 3 ⊢ (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V
13	12	pwex 4320	. 2 ⊢ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V
14		vex 2824	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
15		vex 2824	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
16	14, 15	fvex 5715	. . . . . . . . 9 ⊢ (𝑓‘𝑥) ∈ V
17		vex 2824	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
18	17, 15	fvex 5715	. . . . . . . . 9 ⊢ (𝑔‘𝑥) ∈ V
19		ovssunirng 6120	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) ∈ V ∧ (𝑔‘𝑥) ∈ V) → ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (Hom ‘(𝑟‘𝑥)))
20	16, 18, 19	mp2an 430	. . . . . . . 8 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (Hom ‘(𝑟‘𝑥))
21		homid 13590	. . . . . . . . . . . 12 ⊢ Hom = Slot (Hom ‘ndx)
22	3, 15	fvex 5715	. . . . . . . . . . . . 13 ⊢ (𝑟‘𝑥) ∈ V
23	22	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑟‘𝑥) ∈ V)
24		homslid 13591	. . . . . . . . . . . . . 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 13376	. . . . . . . . . . 11 ⊢ (⊤ → (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥))
28	27	mptru 1411	. . . . . . . . . 10 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥)
29		fvssunirng 5710	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → (𝑟‘𝑥) ⊆ ∪ ran 𝑟)
30	29	elv 2825	. . . . . . . . . . 11 ⊢ (𝑟‘𝑥) ⊆ ∪ ran 𝑟
31		rnss 5012	. . . . . . . . . . 11 ⊢ ((𝑟‘𝑥) ⊆ ∪ ran 𝑟 → ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟)
32		uniss 3956	. . . . . . . . . . 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 3257	. . . . . . . . 9 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟
35		rnss 5012	. . . . . . . . 9 ⊢ ((Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 → ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟)
36		uniss 3956	. . . . . . . . 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 3257	. . . . . . 7 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
39	38	rgenw 2605	. . . . . 6 ⊢ ∀𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
40		ss2ixp 6993	. . . . . 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 6990	. . . . 5 ⊢ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟 = (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
43	41, 42	sseqtri 3282	. . . 4 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
44	12, 43	elpwi2 4294	. . 3 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
45	44	rgen2w 2606	. 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 6449	1 ⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V

This proof depends on syntax axioms:   = wceq 1402  ⊤wtru 1403   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688  ∪ cuni 3935   × cxp 4772  dom cdm 4774  ran crn 4775   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085   ∈ cmpo 6087   ↑_𝑚 cmap 6922  Xcixp 6980  ℕcn 9305  ndxcnx 13351  Slot cslot 13353  Hom chom 13444

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-ixp 6981  df-sub 8499  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-dec 9780  df-ndx 13357  df-slot 13358  df-hom 13457

This theorem is used by:  prdsval  14175