Theorem finxpreclem3 37598

Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.)

Hypothesis

Ref	Expression
finxpreclem3.1	⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))

Assertion

Ref	Expression
finxpreclem3	⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Distinct variable groups:   𝑛,𝑁,𝑥   𝑈,𝑛,𝑥   𝑛,𝑋,𝑥

Allowed substitution hints:   𝐹(𝑥,𝑛)

Proof of Theorem finxpreclem3

Step	Hyp	Ref	Expression
1		finxpreclem3.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
2	1	a1i 11	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))))
3		eqeq1 2740	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 = 1o ↔ 𝑁 = 1o))
4		eleq1 2824	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
5	3, 4	bi2anan9 638	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ((𝑛 = 1o ∧ 𝑥 ∈ 𝑈) ↔ (𝑁 = 1o ∧ 𝑋 ∈ 𝑈)))
6		eleq1 2824	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
7	6	adantl 481	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
8		unieq 4874	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ∪ 𝑛 = ∪ 𝑁)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ∪ 𝑛 = ∪ 𝑁)
10		fveq2 6834	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (1st ‘𝑥) = (1st ‘𝑋))
11	10	adantl 481	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (1st ‘𝑥) = (1st ‘𝑋))
12	9, 11	opeq12d 4837	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨∪ 𝑛, (1st ‘𝑥)⟩ = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
13		opeq12 4831	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
14	7, 12, 13	ifbieq12d 4508	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩))
15	5, 14	ifbieq2d 4506	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if((𝑁 = 1o ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)))
16		sssucid 6399	. . . . . . . . . . . . 13 ⊢ 1o ⊆ suc 1o
17		df-2o 8398	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
18	16, 17	sseqtrri 3983	. . . . . . . . . . . 12 ⊢ 1o ⊆ 2o
19		1on 8409	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
20	17, 19	sucneqoni 37571	. . . . . . . . . . . . 13 ⊢ 2o ≠ 1o
21	20	necomi 2986	. . . . . . . . . . . 12 ⊢ 1o ≠ 2o
22		df-pss 3921	. . . . . . . . . . . 12 ⊢ (1o ⊊ 2o ↔ (1o ⊆ 2o ∧ 1o ≠ 2o))
23	18, 21, 22	mpbir2an 711	. . . . . . . . . . 11 ⊢ 1o ⊊ 2o
24		ssnpss 4058	. . . . . . . . . . 11 ⊢ (2o ⊆ 1o → ¬ 1o ⊊ 2o)
25	23, 24	mt2 200	. . . . . . . . . 10 ⊢ ¬ 2o ⊆ 1o
26		sseq2 3960	. . . . . . . . . 10 ⊢ (𝑁 = 1o → (2o ⊆ 𝑁 ↔ 2o ⊆ 1o))
27	25, 26	mtbiri 327	. . . . . . . . 9 ⊢ (𝑁 = 1o → ¬ 2o ⊆ 𝑁)
28	27	con2i 139	. . . . . . . 8 ⊢ (2o ⊆ 𝑁 → ¬ 𝑁 = 1o)
29	28	intnanrd 489	. . . . . . 7 ⊢ (2o ⊆ 𝑁 → ¬ (𝑁 = 1o ∧ 𝑋 ∈ 𝑈))
30	29	iffalsed 4490	. . . . . 6 ⊢ (2o ⊆ 𝑁 → if((𝑁 = 1o ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩))
31		iftrue 4485	. . . . . 6 ⊢ (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
32	30, 31	sylan9eq 2791	. . . . 5 ⊢ ((2o ⊆ 𝑁 ∧ 𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1o ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
33	15, 32	sylan9eqr 2793	. . . 4 ⊢ (((2o ⊆ 𝑁 ∧ 𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁 ∧ 𝑥 = 𝑋)) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
34	33	adantlll 718	. . 3 ⊢ ((((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁 ∧ 𝑥 = 𝑋)) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
35		simpll 766	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
36		elex 3461	. . . 4 ⊢ (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
37	36	adantl 481	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
38		opex 5412	. . . 4 ⊢ ⟨∪ 𝑁, (1st ‘𝑋)⟩ ∈ V
39	38	a1i 11	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑋)⟩ ∈ V)
40	2, 34, 35, 37, 39	ovmpod 7510	. 2 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨∪ 𝑁, (1st ‘𝑋)⟩)
41		df-ov 7361	. 2 ⊢ (𝑁𝐹𝑋) = (𝐹‘⟨𝑁, 𝑋⟩)
42	40, 41	eqtr3di 2786	1 ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440   ⊆ wss 3901   ⊊ wpss 3902  ∅c0 4285  ifcif 4479  ⟨cop 4586  ∪ cuni 4863   × cxp 5622  suc csuc 6319  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  ωcom 7808  1^st c1st 7931  1_oc1o 8390  2_oc2o 8391

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-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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-rab 3400  df-v 3442  df-sbc 3741  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-br 5099  df-opab 5161  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-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1o 8397  df-2o 8398

This theorem is referenced by:  finxpreclem4  37599