Theorem fin23lem16 10292

Description: Lemma for fin23 10346. 𝑈 ranges over the original set; in particular ran 𝑈 is a set, although we do not assume here that 𝑈 is. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Hypothesis

Ref	Expression
fin23lem.a	⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)

Assertion

Ref	Expression
fin23lem16	⊢ ∪ ran 𝑈 = ∪ ran 𝑡

Distinct variable groups:   𝑡,𝑖,𝑢   𝑈,𝑖,𝑢

Allowed substitution hint:   𝑈(𝑡)

Proof of Theorem fin23lem16

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unissb 4899	. . 3 ⊢ (∪ ran 𝑈 ⊆ ∪ ran 𝑡 ↔ ∀𝑎 ∈ ran 𝑈 𝑎 ⊆ ∪ ran 𝑡)
2		fin23lem.a	. . . . . 6 ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
3	2	fnseqom 8426	. . . . 5 ⊢ 𝑈 Fn ω
4		fvelrnb 6927	. . . . 5 ⊢ (𝑈 Fn ω → (𝑎 ∈ ran 𝑈 ↔ ∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎))
5	3, 4	ax-mp 5	. . . 4 ⊢ (𝑎 ∈ ran 𝑈 ↔ ∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎)
6		peano1 7869	. . . . . . . 8 ⊢ ∅ ∈ ω
7		0ss 4354	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑏
8	2	fin23lem15 10291	. . . . . . . . 9 ⊢ (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈‘𝑏) ⊆ (𝑈‘∅))
9	7, 8	mpan2 701	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈‘𝑏) ⊆ (𝑈‘∅))
10	6, 9	mpan2 701	. . . . . . 7 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ (𝑈‘∅))
11		vex 3458	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
12	11	rnex 7891	. . . . . . . . 9 ⊢ ran 𝑡 ∈ V
13	12	uniex 7724	. . . . . . . 8 ⊢ ∪ ran 𝑡 ∈ V
14	2	seqom0g 8427	. . . . . . . 8 ⊢ (∪ ran 𝑡 ∈ V → (𝑈‘∅) = ∪ ran 𝑡)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (𝑈‘∅) = ∪ ran 𝑡
16	10, 15	sseqtrdi 3976	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ ∪ ran 𝑡)
17		sseq1 3961	. . . . . 6 ⊢ ((𝑈‘𝑏) = 𝑎 → ((𝑈‘𝑏) ⊆ ∪ ran 𝑡 ↔ 𝑎 ⊆ ∪ ran 𝑡))
18	16, 17	syl5ibcom 247	. . . . 5 ⊢ (𝑏 ∈ ω → ((𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡))
19	18	rexlimiv 3156	. . . 4 ⊢ (∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡)
20	5, 19	sylbi 219	. . 3 ⊢ (𝑎 ∈ ran 𝑈 → 𝑎 ⊆ ∪ ran 𝑡)
21	1, 20	mprgbir 3083	. 2 ⊢ ∪ ran 𝑈 ⊆ ∪ ran 𝑡
22		fnfvelrn 7061	. . . . 5 ⊢ ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈)
23	3, 6, 22	mp2an 702	. . . 4 ⊢ (𝑈‘∅) ∈ ran 𝑈
24	15, 23	eqeltrri 2859	. . 3 ⊢ ∪ ran 𝑡 ∈ ran 𝑈
25		elssuni 4897	. . 3 ⊢ (∪ ran 𝑡 ∈ ran 𝑈 → ∪ ran 𝑡 ⊆ ∪ ran 𝑈)
26	24, 25	ax-mp 5	. 2 ⊢ ∪ ran 𝑡 ⊆ ∪ ran 𝑈
27	21, 26	eqssi 3952	1 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∃wrex 3086  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ifcif 4480  ∪ cuni 4865  ran crn 5648   Fn wfn 6516  ‘cfv 6521   ∈ cmpo 7398  ωcom 7846  seq_ωcseqom 8418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-seqom 8419

This theorem is referenced by:  fin23lem17  10295  fin23lem31  10300