Theorem xpsfrnel 17609

Description: Elementhood in the target space of the function 𝐹 appearing in xpsval 17617. (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 8940	. 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 8679	. . . . . . . . . 10 ⊢ 2o ∈ ω
4		nnfi 9206	. . . . . . . . . 10 ⊢ (2o ∈ ω → 2o ∈ Fin)
5	3, 4	ax-mp 5	. . . . . . . . 9 ⊢ 2o ∈ Fin
6		fnfi 9216	. . . . . . . . 9 ⊢ ((𝐺 Fn 2o ∧ 2o ∈ Fin) → 𝐺 ∈ Fin)
7	5, 6	mpan2 691	. . . . . . . 8 ⊢ (𝐺 Fn 2o → 𝐺 ∈ Fin)
8	7	elexd 3502	. . . . . . 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 8513	. . . . . . . 8 ⊢ 2o = {∅, 1o}
11	10	raleqi 3322	. . . . . . 7 ⊢ (∀𝑘 ∈ 2o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ∀𝑘 ∈ {∅, 1o} (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))
12		0ex 5313	. . . . . . . 8 ⊢ ∅ ∈ V
13		1oex 8515	. . . . . . . 8 ⊢ 1o ∈ V
14		fveq2 6907	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
15		iftrue 4537	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
16	14, 15	eleq12d 2833	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺‘∅) ∈ 𝐴))
17		fveq2 6907	. . . . . . . . 9 ⊢ (𝑘 = 1o → (𝐺‘𝑘) = (𝐺‘1o))
18		1n0 8525	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
19		neeq1 3001	. . . . . . . . . . 11 ⊢ (𝑘 = 1o → (𝑘 ≠ ∅ ↔ 1o ≠ ∅))
20	18, 19	mpbiri 258	. . . . . . . . . 10 ⊢ (𝑘 = 1o → 𝑘 ≠ ∅)
21		ifnefalse 4543	. . . . . . . . . 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 4705	. . . . . . 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 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  Vcvv 3478  ∅c0 4339  ifcif 4531  {cpr 4633   Fn wfn 6558  ‘cfv 6563  ωcom 7887  1_oc1o 8498  2_oc2o 8499  Xcixp 8936  Fincfn 8984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506  df-ixp 8937  df-en 8985  df-fin 8988

This theorem is referenced by:  xpsfrnel2  17611  xpsff1o  17614