Theorem xpsfrnel 17190

Description: Elementhood in the target space of the function 𝐹 appearing in xpsval 17198. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
xpsfrnel	⊢ (𝐺 ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵))

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑘,𝐺

Proof of Theorem xpsfrnel

Step	Hyp	Ref	Expression
1		elixp2 8647	. 2 ⊢ (𝐺 ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 ∈ V ∧ 𝐺 Fn 2o ∧ ∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)))
2		3ancoma 1096	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐺 Fn 2o ∧ ∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ (𝐺 Fn 2o ∧ 𝐺 ∈ V ∧ ∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)))
3		2onn 8433	. . . . . . . . . 10 ⊢ 2o ∈ ω
4		nnfi 8912	. . . . . . . . . 10 ⊢ (2o ∈ ω → 2o ∈ Fin)
5	3, 4	ax-mp 5	. . . . . . . . 9 ⊢ 2o ∈ Fin
6		fnfi 8925	. . . . . . . . 9 ⊢ ((𝐺 Fn 2o ∧ 2o ∈ Fin) → 𝐺 ∈ Fin)
7	5, 6	mpan2 687	. . . . . . . 8 ⊢ (𝐺 Fn 2o → 𝐺 ∈ Fin)
8	7	elexd 3442	. . . . . . 7 ⊢ (𝐺 Fn 2o → 𝐺 ∈ V)
9	8	biantrurd 532	. . . . . 6 ⊢ (𝐺 Fn 2o → (∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 ∈ V ∧ ∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))))
10		df2o3 8282	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	10	raleqi 3337	. . . . . . 7 ⊢ (∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ∀𝑘 ∈ {∅, 1o} (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))
12		0ex 5226	. . . . . . . 8 ⊢ ∅ ∈ V
13		1oex 8280	. . . . . . . 8 ⊢ 1o ∈ V
14		fveq2 6756	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
15		iftrue 4462	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
16	14, 15	eleq12d 2833	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺‘∅) ∈ 𝐴))
17		fveq2 6756	. . . . . . . . 9 ⊢ (𝑘 = 1o → (𝐺‘𝑘) = (𝐺‘1o))
18		1n0 8286	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
19		neeq1 3005	. . . . . . . . . . 11 ⊢ (𝑘 = 1o → (𝑘 ≠ ∅ ↔ 1o ≠ ∅))
20	18, 19	mpbiri 257	. . . . . . . . . 10 ⊢ (𝑘 = 1o → 𝑘 ≠ ∅)
21		ifnefalse 4468	. . . . . . . . . 10 ⊢ (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
22	20, 21	syl 17	. . . . . . . . 9 ⊢ (𝑘 = 1o → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
23	17, 22	eleq12d 2833	. . . . . . . 8 ⊢ (𝑘 = 1o → ((𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺‘1o) ∈ 𝐵))
24	12, 13, 16, 23	ralpr 4633	. . . . . . 7 ⊢ (∀𝑘 ∈ {∅, 1o} (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵))
25	11, 24	bitri 274	. . . . . 6 ⊢ (∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵))
26	9, 25	bitr3di 285	. . . . 5 ⊢ (𝐺 Fn 2o → ((𝐺 ∈ V ∧ ∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵)))
27	26	pm5.32i 574	. . . 4 ⊢ ((𝐺 Fn 2o ∧ (𝐺 ∈ V ∧ ∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))) ↔ (𝐺 Fn 2o ∧ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵)))
28		3anass 1093	. . . 4 ⊢ ((𝐺 Fn 2o ∧ 𝐺 ∈ V ∧ ∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ (𝐺 Fn 2o ∧ (𝐺 ∈ V ∧ ∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))))
29		3anass 1093	. . . 4 ⊢ ((𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵) ↔ (𝐺 Fn 2o ∧ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵)))
30	27, 28, 29	3bitr4i 302	. . 3 ⊢ ((𝐺 Fn 2o ∧ 𝐺 ∈ V ∧ ∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ (𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵))
31	2, 30	bitri 274	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐺 Fn 2o ∧ ∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ (𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵))
32	1, 31	bitri 274	1 ⊢ (𝐺 ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422  ∅c0 4253  ifcif 4456  {cpr 4560   Fn wfn 6413  ‘cfv 6418  ωcom 7687  1_oc1o 8260  2_oc2o 8261  Xcixp 8643  Fincfn 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-2o 8268  df-ixp 8644  df-en 8692  df-fin 8695

This theorem is referenced by:  xpsfrnel2  17192  xpsff1o  17195