Theorem xpsfrnel 16046

Description: Elementhood in the target space of the function 𝐹 appearing in xpsval 16055. (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 7798	. 2 ⊢ (𝐺 ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 ∈ V ∧ 𝐺 Fn 2𝑜 ∧ ∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)))
2		3ancoma 1038	. . 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		df2o3 7460	. . . . . . . 8 ⊢ 2𝑜 = {∅, 1𝑜}
4	3	raleqi 3119	. . . . . . 7 ⊢ (∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ∀𝑘 ∈ {∅, 1𝑜} (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))
5		0ex 4718	. . . . . . . 8 ⊢ ∅ ∈ V
6		1on 7454	. . . . . . . . 9 ⊢ 1𝑜 ∈ On
7	6	elexi 3186	. . . . . . . 8 ⊢ 1𝑜 ∈ V
8		fveq2 6103	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
9		iftrue 4042	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
10	8, 9	eleq12d 2682	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺‘∅) ∈ 𝐴))
11		fveq2 6103	. . . . . . . . 9 ⊢ (𝑘 = 1𝑜 → (𝐺‘𝑘) = (𝐺‘1𝑜))
12		1n0 7462	. . . . . . . . . . 11 ⊢ 1𝑜 ≠ ∅
13		neeq1 2844	. . . . . . . . . . 11 ⊢ (𝑘 = 1𝑜 → (𝑘 ≠ ∅ ↔ 1𝑜 ≠ ∅))
14	12, 13	mpbiri 247	. . . . . . . . . 10 ⊢ (𝑘 = 1𝑜 → 𝑘 ≠ ∅)
15		ifnefalse 4048	. . . . . . . . . 10 ⊢ (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑘 = 1𝑜 → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
17	11, 16	eleq12d 2682	. . . . . . . 8 ⊢ (𝑘 = 1𝑜 → ((𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺‘1𝑜) ∈ 𝐵))
18	5, 7, 10, 17	ralpr 4185	. . . . . . 7 ⊢ (∀𝑘 ∈ {∅, 1𝑜} (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵))
19	4, 18	bitri 263	. . . . . 6 ⊢ (∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵))
20		2onn 7607	. . . . . . . . . 10 ⊢ 2𝑜 ∈ ω
21		nnfi 8038	. . . . . . . . . 10 ⊢ (2𝑜 ∈ ω → 2𝑜 ∈ Fin)
22	20, 21	ax-mp 5	. . . . . . . . 9 ⊢ 2𝑜 ∈ Fin
23		fnfi 8123	. . . . . . . . 9 ⊢ ((𝐺 Fn 2𝑜 ∧ 2𝑜 ∈ Fin) → 𝐺 ∈ Fin)
24	22, 23	mpan2 703	. . . . . . . 8 ⊢ (𝐺 Fn 2𝑜 → 𝐺 ∈ Fin)
25		elex 3185	. . . . . . . 8 ⊢ (𝐺 ∈ Fin → 𝐺 ∈ V)
26	24, 25	syl 17	. . . . . . 7 ⊢ (𝐺 Fn 2𝑜 → 𝐺 ∈ V)
27	26	biantrurd 528	. . . . . 6 ⊢ (𝐺 Fn 2𝑜 → (∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 ∈ V ∧ ∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))))
28	19, 27	syl5rbbr 274	. . . . 5 ⊢ (𝐺 Fn 2𝑜 → ((𝐺 ∈ V ∧ ∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵)))
29	28	pm5.32i 667	. . . 4 ⊢ ((𝐺 Fn 2𝑜 ∧ (𝐺 ∈ V ∧ ∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))) ↔ (𝐺 Fn 2𝑜 ∧ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵)))
30		3anass 1035	. . . 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))))
31		3anass 1035	. . . 4 ⊢ ((𝐺 Fn 2𝑜 ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵) ↔ (𝐺 Fn 2𝑜 ∧ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵)))
32	29, 30, 31	3bitr4i 291	. . 3 ⊢ ((𝐺 Fn 2𝑜 ∧ 𝐺 ∈ V ∧ ∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ (𝐺 Fn 2𝑜 ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵))
33	2, 32	bitri 263	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐺 Fn 2𝑜 ∧ ∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ (𝐺 Fn 2𝑜 ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵))
34	1, 33	bitri 263	1 ⊢ (𝐺 ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2𝑜 ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵))

Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874  ifcif 4036  {cpr 4127  Oncon0 5640   Fn wfn 5799  ‘cfv 5804  ωcom 6957  1_𝑜c1o 7440  2_𝑜c2o 7441  Xcixp 7794  Fincfn 7841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845

This theorem is referenced by:  xpsfrnel2  16048  xpsff1o  16051