Theorem itunisuc 10329

Description: Successor iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)

Hypothesis

Ref	Expression
ituni.u	⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))

Assertion

Ref	Expression
itunisuc	⊢ ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝑈(𝑥,𝑦)

Proof of Theorem itunisuc

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frsuc 8368	. . . . . 6 ⊢ (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑦 ∈ V ↦ ∪ 𝑦)‘((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵)))
2		fvex 6847	. . . . . . 7 ⊢ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
3		unieq 4874	. . . . . . . 8 ⊢ (𝑎 = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) → ∪ 𝑎 = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
4		unieq 4874	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → ∪ 𝑦 = ∪ 𝑎)
5	4	cbvmptv 5202	. . . . . . . 8 ⊢ (𝑦 ∈ V ↦ ∪ 𝑦) = (𝑎 ∈ V ↦ ∪ 𝑎)
6	2	uniex 7686	. . . . . . . 8 ⊢ ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
7	3, 5, 6	fvmpt 6941	. . . . . . 7 ⊢ (((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V → ((𝑦 ∈ V ↦ ∪ 𝑦)‘((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵)) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
8	2, 7	ax-mp 5	. . . . . 6 ⊢ ((𝑦 ∈ V ↦ ∪ 𝑦)‘((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵)) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵)
9	1, 8	eqtrdi 2787	. . . . 5 ⊢ (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
10	9	adantl 481	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
11		ituni.u	. . . . . . 7 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
12	11	itunifval 10326	. . . . . 6 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω))
13	12	fveq1d 6836	. . . . 5 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
14	13	adantr 480	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
15	12	fveq1d 6836	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
16	15	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
17	16	unieqd 4876	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
18	10, 14, 17	3eqtr4d 2781	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
19		uni0 4891	. . . . 5 ⊢ ∪ ∅ = ∅
20	19	eqcomi 2745	. . . 4 ⊢ ∅ = ∪ ∅
21	11	itunifn 10327	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) Fn ω)
22	21	fndmd 6597	. . . . . . . . 9 ⊢ (𝐴 ∈ V → dom (𝑈‘𝐴) = ω)
23	22	eleq2d 2822	. . . . . . . 8 ⊢ (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ suc 𝐵 ∈ ω))
24		peano2b 7825	. . . . . . . 8 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
25	23, 24	bitr4di 289	. . . . . . 7 ⊢ (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ 𝐵 ∈ ω))
26	25	notbid 318	. . . . . 6 ⊢ (𝐴 ∈ V → (¬ suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ ¬ 𝐵 ∈ ω))
27	26	biimpar 477	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ suc 𝐵 ∈ dom (𝑈‘𝐴))
28		ndmfv 6866	. . . . 5 ⊢ (¬ suc 𝐵 ∈ dom (𝑈‘𝐴) → ((𝑈‘𝐴)‘suc 𝐵) = ∅)
29	27, 28	syl 17	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∅)
30	22	eleq2d 2822	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐵 ∈ dom (𝑈‘𝐴) ↔ 𝐵 ∈ ω))
31	30	notbid 318	. . . . . . 7 ⊢ (𝐴 ∈ V → (¬ 𝐵 ∈ dom (𝑈‘𝐴) ↔ ¬ 𝐵 ∈ ω))
32	31	biimpar 477	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ 𝐵 ∈ dom (𝑈‘𝐴))
33		ndmfv 6866	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝐴) → ((𝑈‘𝐴)‘𝐵) = ∅)
34	32, 33	syl 17	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘𝐵) = ∅)
35	34	unieqd 4876	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ ∅)
36	20, 29, 35	3eqtr4a 2797	. . 3 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
37	18, 36	pm2.61dan 812	. 2 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
38		0fv 6875	. . . . 5 ⊢ (∅‘𝐵) = ∅
39	38	unieqi 4875	. . . 4 ⊢ ∪ (∅‘𝐵) = ∪ ∅
40		0fv 6875	. . . 4 ⊢ (∅‘suc 𝐵) = ∅
41	19, 39, 40	3eqtr4ri 2770	. . 3 ⊢ (∅‘suc 𝐵) = ∪ (∅‘𝐵)
42		fvprc 6826	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑈‘𝐴) = ∅)
43	42	fveq1d 6836	. . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = (∅‘suc 𝐵))
44	42	fveq1d 6836	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = (∅‘𝐵))
45	44	unieqd 4876	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ (∅‘𝐵))
46	41, 43, 45	3eqtr4a 2797	. 2 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
47	37, 46	pm2.61i 182	1 ⊢ ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  ∪ cuni 4863   ↦ cmpt 5179  dom cdm 5624   ↾ cres 5626  suc csuc 6319  ‘cfv 6492  ωcom 7808  reccrdg 8340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341

This theorem is referenced by:  itunitc1  10330  itunitc  10331  ituniiun  10332