Theorem prdsvallem 17539

Description: Lemma for prdsval 17540. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 17540, dependency on df-hom 17366 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 3454	. 2 ⊢ 𝑣 ∈ V
2		ovex 7446	. . 3 ⊢ (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) ∈ V
3	2	pwex 5345	. 2 ⊢ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟) ∈ V
4		ovssunirn 7449	. . . . . . . 8 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran (Hom ‘(𝑟‘𝑥))
5		homid 17497	. . . . . . . . . . 11 ⊢ Hom = Slot (Hom ‘ndx)
6	5	strfvss 17279	. . . . . . . . . 10 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥)
7		fvssunirn 6909	. . . . . . . . . . 11 ⊢ (𝑟‘𝑥) ⊆ ∪ ran 𝑟
8		rnss 5923	. . . . . . . . . . 11 ⊢ ((𝑟‘𝑥) ⊆ ∪ ran 𝑟 → ran (𝑟‘𝑥) ⊆ ran ∪ ran 𝑟)
9		uniss 4875	. . . . . . . . . . 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 3940	. . . . . . . . 9 ⊢ (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟
12		rnss 5923	. . . . . . . . 9 ⊢ ((Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 → ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟)
13		uniss 4875	. . . . . . . . 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 3940	. . . . . . 7 ⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
16	15	rgenw 3080	. . . . . 6 ⊢ ∀𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟
17		ss2ixp 8917	. . . . . 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 3454	. . . . . . 7 ⊢ 𝑟 ∈ V
20	19	dmex 7906	. . . . . 6 ⊢ dom 𝑟 ∈ V
21	19	rnex 7907	. . . . . . . . . . 11 ⊢ ran 𝑟 ∈ V
22	21	uniex 7743	. . . . . . . . . 10 ⊢ ∪ ran 𝑟 ∈ V
23	22	rnex 7907	. . . . . . . . 9 ⊢ ran ∪ ran 𝑟 ∈ V
24	23	uniex 7743	. . . . . . . 8 ⊢ ∪ ran ∪ ran 𝑟 ∈ V
25	24	rnex 7907	. . . . . . 7 ⊢ ran ∪ ran ∪ ran 𝑟 ∈ V
26	25	uniex 7743	. . . . . 6 ⊢ ∪ ran ∪ ran ∪ ran 𝑟 ∈ V
27	20, 26	ixpconst 8914	. . . . 5 ⊢ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟 = (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟)
28	18, 27	sseqtri 3979	. . . 4 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟)
29	2, 28	elpwi2 5300	. . 3 ⊢ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑m dom 𝑟)
30	29	rgen2w 3081	. 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 8077	1 ⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V

This proof depends on syntax axioms:   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  dom cdm 5655  ran crn 5656  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415   ↑_m cmap 8826  Xcixp 8904  ndxcnx 17285  Hom chom 17353

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-map 8828  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11269  df-mnf 11270  df-ltxr 11272  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-dec 12737  df-slot 17274  df-ndx 17286  df-hom 17366

This theorem is used by:  prdsval  17540