Theorem prdsvallem 13604

Description: Lemma for prdsval 14156. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 14156, dependency on df-hom 13438 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 6923	. . . 4 ⊢ ↑𝑚 Fn (V × V)
3		vex 2824	. . . . . . . . . 10 ⊢ 𝑟 ∈ V
4	3	rnex 5048	. . . . . . . . 9 ⊢ ran 𝑟 ∈ V
5	4	uniex 4581	. . . . . . . 8 ⊢ ∪ ran 𝑟 ∈ V
6	5	rnex 5048	. . . . . . 7 ⊢ ran ∪ ran 𝑟 ∈ V
7	6	uniex 4581	. . . . . 6 ⊢ ∪ ran ∪ ran 𝑟 ∈ V
8	7	rnex 5048	. . . . 5 ⊢ ran ∪ ran ∪ ran 𝑟 ∈ V
9	8	uniex 4581	. . . 4 ⊢ ∪ ran ∪ ran ∪ ran 𝑟 ∈ V
10	3	dmex 5047	. . . 4 ⊢ dom 𝑟 ∈ V
11		fnovex 6112	. . . 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 4318	. 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 5713	. . . . . . . . 9 ⊢ (𝑓‘𝑥) ∈ V
17		vex 2824	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
18	17, 15	fvex 5713	. . . . . . . . 9 ⊢ (𝑔‘𝑥) ∈ V
19		ovssunirng 6114	. . . . . . . . 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 13571	. . . . . . . . . . . 12 ⊢ Hom = Slot (Hom ‘ndx)
22	3, 15	fvex 5713	. . . . . . . . . . . . 13 ⊢ (𝑟‘𝑥) ∈ V
23	22	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑟‘𝑥) ∈ V)
24		homslid 13572	. . . . . . . . . . . . . 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 13357	. . . . . . . . . . 11 ⊢ (⊤ → (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥))
28	27	mptru 1411	. . . . . . . . . 10 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥)
29		fvssunirng 5708	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → (𝑟‘𝑥) ⊆ ∪ ran 𝑟)
30	29	elv 2825	. . . . . . . . . . 11 ⊢ (𝑟‘𝑥) ⊆ ∪ ran 𝑟
31		rnss 5010	. . . . . . . . . . 11 ⊢ ((𝑟‘𝑥) ⊆ ∪ ran 𝑟 → ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟)
32		uniss 3954	. . . . . . . . . . 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 5010	. . . . . . . . 9 ⊢ ((Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 → ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟)
36		uniss 3954	. . . . . . . . 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 6987	. . . . . 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 6984	. . . . 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 4292	. . 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 6443	1 ⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V

Syntax hints:   = wceq 1402  ⊤wtru 1403   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688  ∪ cuni 3933   × cxp 4770  dom cdm 4772  ran crn 4773   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079   ∈ cmpo 6081   ↑_𝑚 cmap 6916  Xcixp 6974  ℕcn 9287  ndxcnx 13332  Slot cslot 13334  Hom chom 13425

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 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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-map 6918  df-ixp 6975  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-hom 13438

This theorem is referenced by:  prdsval  14156