Theorem prdsvallem 13271

Description: Lemma for prdsval 13272. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 13272, dependency on df-hom 13100 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 2782	. 2 ⊢ 𝑣 ∈ V
2		fnmap 6772	. . . 4 ⊢ ↑𝑚 Fn (V × V)
3		vex 2782	. . . . . . . . . 10 ⊢ 𝑟 ∈ V
4	3	rnex 4968	. . . . . . . . 9 ⊢ ran 𝑟 ∈ V
5	4	uniex 4505	. . . . . . . 8 ⊢ ∪ ran 𝑟 ∈ V
6	5	rnex 4968	. . . . . . 7 ⊢ ran ∪ ran 𝑟 ∈ V
7	6	uniex 4505	. . . . . 6 ⊢ ∪ ran ∪ ran 𝑟 ∈ V
8	7	rnex 4968	. . . . 5 ⊢ ran ∪ ran ∪ ran 𝑟 ∈ V
9	8	uniex 4505	. . . 4 ⊢ ∪ ran ∪ ran ∪ ran 𝑟 ∈ V
10	3	dmex 4967	. . . 4 ⊢ dom 𝑟 ∈ V
11		fnovex 6007	. . . 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 1352	. . 3 ⊢ (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V
13	12	pwex 4246	. 2 ⊢ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V
14		vex 2782	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
15		vex 2782	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
16	14, 15	fvex 5623	. . . . . . . . 9 ⊢ (𝑓‘𝑥) ∈ V
17		vex 2782	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
18	17, 15	fvex 5623	. . . . . . . . 9 ⊢ (𝑔‘𝑥) ∈ V
19		ovssunirng 6009	. . . . . . . . 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 13233	. . . . . . . . . . . 12 ⊢ Hom = Slot (Hom ‘ndx)
22	3, 15	fvex 5623	. . . . . . . . . . . . 13 ⊢ (𝑟‘𝑥) ∈ V
23	22	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑟‘𝑥) ∈ V)
24		homslid 13234	. . . . . . . . . . . . . 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 13020	. . . . . . . . . . 11 ⊢ (⊤ → (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥))
28	27	mptru 1384	. . . . . . . . . 10 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥)
29		fvssunirng 5618	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → (𝑟‘𝑥) ⊆ ∪ ran 𝑟)
30	29	elv 2783	. . . . . . . . . . 11 ⊢ (𝑟‘𝑥) ⊆ ∪ ran 𝑟
31		rnss 4930	. . . . . . . . . . 11 ⊢ ((𝑟‘𝑥) ⊆ ∪ ran 𝑟 → ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟)
32		uniss 3888	. . . . . . . . . . 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 3213	. . . . . . . . 9 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟
35		rnss 4930	. . . . . . . . 9 ⊢ ((Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 → ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟)
36		uniss 3888	. . . . . . . . 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 3213	. . . . . . 7 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
39	38	rgenw 2565	. . . . . 6 ⊢ ∀𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
40		ss2ixp 6828	. . . . . 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 6825	. . . . 5 ⊢ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟 = (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
43	41, 42	sseqtri 3238	. . . 4 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
44	12, 43	elpwi2 4221	. . 3 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)
45	44	rgen2w 2566	. 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 6329	1 ⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V

Syntax hints:   = wceq 1375  ⊤wtru 1376   ∈ wcel 2180  ∀wral 2488  Vcvv 2779   ⊆ wss 3177  𝒫 cpw 3629  ∪ cuni 3867   × cxp 4694  dom cdm 4696  ran crn 4697   Fn wfn 5289  ‘cfv 5294  (class class class)co 5974   ∈ cmpo 5976   ↑_𝑚 cmap 6765  Xcixp 6815  ℕcn 9078  ndxcnx 12995  Slot cslot 12997  Hom chom 13087

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-map 6767  df-ixp 6816  df-sub 8287  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-dec 9547  df-ndx 13001  df-slot 13002  df-hom 13100

This theorem is referenced by:  prdsval  13272