Theorem xpsfrnel 17466

Description: Elementhood in the target space of the function 𝐹 appearing in xpsval 17474. (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 8825	. 2 ⊢ (𝐺 ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 ∈ V ∧ 𝐺 Fn 2o ∧ ∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)))
2		3ancoma 1097	. . 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 8557	. . . . . . . . . 10 ⊢ 2o ∈ ω
4		nnfi 9077	. . . . . . . . . 10 ⊢ (2o ∈ ω → 2o ∈ Fin)
5	3, 4	ax-mp 5	. . . . . . . . 9 ⊢ 2o ∈ Fin
6		fnfi 9087	. . . . . . . . 9 ⊢ ((𝐺 Fn 2o ∧ 2o ∈ Fin) → 𝐺 ∈ Fin)
7	5, 6	mpan2 691	. . . . . . . 8 ⊢ (𝐺 Fn 2o → 𝐺 ∈ Fin)
8	7	elexd 3460	. . . . . . 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 8393	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	10	raleqi 3290	. . . . . . 7 ⊢ (∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ∀𝑘 ∈ {∅, 1o} (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))
12		0ex 5243	. . . . . . . 8 ⊢ ∅ ∈ V
13		1oex 8395	. . . . . . . 8 ⊢ 1o ∈ V
14		fveq2 6822	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
15		iftrue 4478	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
16	14, 15	eleq12d 2825	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺‘∅) ∈ 𝐴))
17		fveq2 6822	. . . . . . . . 9 ⊢ (𝑘 = 1o → (𝐺‘𝑘) = (𝐺‘1o))
18		1n0 8403	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
19		neeq1 2990	. . . . . . . . . . 11 ⊢ (𝑘 = 1o → (𝑘 ≠ ∅ ↔ 1o ≠ ∅))
20	18, 19	mpbiri 258	. . . . . . . . . 10 ⊢ (𝑘 = 1o → 𝑘 ≠ ∅)
21		ifnefalse 4484	. . . . . . . . . 10 ⊢ (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
22	20, 21	syl 17	. . . . . . . . 9 ⊢ (𝑘 = 1o → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
23	17, 22	eleq12d 2825	. . . . . . . 8 ⊢ (𝑘 = 1o → ((𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺‘1o) ∈ 𝐵))
24	12, 13, 16, 23	ralpr 4650	. . . . . . 7 ⊢ (∀𝑘 ∈ {∅, 1o} (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵))
25	11, 24	bitri 275	. . . . . 6 ⊢ (∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵))
26	9, 25	bitr3di 286	. . . . 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 1094	. . . 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 1094	. . . 4 ⊢ ((𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵) ↔ (𝐺 Fn 2o ∧ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵)))
30	27, 28, 29	3bitr4i 303	. . 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 275	. 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 275	1 ⊢ (𝐺 ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1o) ∈ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  Vcvv 3436  ∅c0 4280  ifcif 4472  {cpr 4575   Fn wfn 6476  ‘cfv 6481  ωcom 7796  1_oc1o 8378  2_oc2o 8379  Xcixp 8821  Fincfn 8869

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-1o 8385  df-2o 8386  df-ixp 8822  df-en 8870  df-fin 8873

This theorem is referenced by:  xpsfrnel2  17468  xpsff1o  17471