Theorem prdsvallem 17417

Description: Lemma for prdsval 17418. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 17418, dependency on df-hom 17244 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 3451	. 2 ⊢ 𝑣 ∈ V
2		ovex 7420	. . 3 ⊢ (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) ∈ V
3	2	pwex 5335	. 2 ⊢ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) ∈ V
4		ovssunirn 7423	. . . . . . . 8 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (Hom ‘(𝑟‘𝑥))
5		homid 17375	. . . . . . . . . . 11 ⊢ Hom = Slot (Hom ‘ndx)
6	5	strfvss 17157	. . . . . . . . . 10 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥)
7		fvssunirn 6891	. . . . . . . . . . 11 ⊢ (𝑟‘𝑥) ⊆ ∪ ran 𝑟
8		rnss 5903	. . . . . . . . . . 11 ⊢ ((𝑟‘𝑥) ⊆ ∪ ran 𝑟 → ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟)
9		uniss 4879	. . . . . . . . . . 11 ⊢ (ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟 → ∪ ran (𝑟‘𝑥) ⊆ ∪ ran ∪ ran 𝑟)
10	7, 8, 9	mp2b 10	. . . . . . . . . 10 ⊢ ∪ ran (𝑟‘𝑥) ⊆ ∪ ran ∪ ran 𝑟
11	6, 10	sstri 3956	. . . . . . . . 9 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟
12		rnss 5903	. . . . . . . . 9 ⊢ ((Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 → ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟)
13		uniss 4879	. . . . . . . . 9 ⊢ (ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟 → ∪ ran (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟)
14	11, 12, 13	mp2b 10	. . . . . . . 8 ⊢ ∪ ran (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
15	4, 14	sstri 3956	. . . . . . 7 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
16	15	rgenw 3048	. . . . . 6 ⊢ ∀𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
17		ss2ixp 8883	. . . . . 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)
18	16, 17	ax-mp 5	. . . . 5 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟
19		vex 3451	. . . . . . 7 ⊢ 𝑟 ∈ V
20	19	dmex 7885	. . . . . 6 ⊢ dom 𝑟 ∈ V
21	19	rnex 7886	. . . . . . . . . . 11 ⊢ ran 𝑟 ∈ V
22	21	uniex 7717	. . . . . . . . . 10 ⊢ ∪ ran 𝑟 ∈ V
23	22	rnex 7886	. . . . . . . . 9 ⊢ ran ∪ ran 𝑟 ∈ V
24	23	uniex 7717	. . . . . . . 8 ⊢ ∪ ran ∪ ran 𝑟 ∈ V
25	24	rnex 7886	. . . . . . 7 ⊢ ran ∪ ran ∪ ran 𝑟 ∈ V
26	25	uniex 7717	. . . . . 6 ⊢ ∪ ran ∪ ran ∪ ran 𝑟 ∈ V
27	20, 26	ixpconst 8880	. . . . 5 ⊢ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟 = (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟)
28	18, 27	sseqtri 3995	. . . 4 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟)
29	2, 28	elpwi2 5290	. . 3 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟)
30	29	rgen2w 3049	. 2 ⊢ ∀𝑓 ∈ 𝑣 ∀𝑔 ∈ 𝑣 X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟)
31	1, 1, 3, 30	mpoexw 8057	1 ⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V

Syntax hints:   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871  dom cdm 5638  ran crn 5639  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389   ↑_m cmap 8799  Xcixp 8870  ndxcnx 17163  Hom chom 17231

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-ltxr 11213  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-dec 12650  df-slot 17152  df-ndx 17164  df-hom 17244

This theorem is referenced by:  prdsval  17418