Theorem xpsfrnel 16223

Description: Elementhood in the target space of the function 𝐹 appearing in xpsval 16232. (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 7912	. 2 ⊢ (𝐺 ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 ∈ V ∧ 𝐺 Fn 2𝑜 ∧ ∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)))
2		3ancoma 1045	. . 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 7573	. . . . . . . 8 ⊢ 2𝑜 = {∅, 1𝑜}
4	3	raleqi 3142	. . . . . . 7 ⊢ (∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ∀𝑘 ∈ {∅, 1𝑜} (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))
5		0ex 4790	. . . . . . . 8 ⊢ ∅ ∈ V
6		1on 7567	. . . . . . . . 9 ⊢ 1𝑜 ∈ On
7	6	elexi 3213	. . . . . . . 8 ⊢ 1𝑜 ∈ V
8		fveq2 6191	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
9		iftrue 4092	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
10	8, 9	eleq12d 2695	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺‘∅) ∈ 𝐴))
11		fveq2 6191	. . . . . . . . 9 ⊢ (𝑘 = 1𝑜 → (𝐺‘𝑘) = (𝐺‘1𝑜))
12		1n0 7575	. . . . . . . . . . 11 ⊢ 1𝑜 ≠ ∅
13		neeq1 2856	. . . . . . . . . . 11 ⊢ (𝑘 = 1𝑜 → (𝑘 ≠ ∅ ↔ 1𝑜 ≠ ∅))
14	12, 13	mpbiri 248	. . . . . . . . . 10 ⊢ (𝑘 = 1𝑜 → 𝑘 ≠ ∅)
15		ifnefalse 4098	. . . . . . . . . 10 ⊢ (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑘 = 1𝑜 → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
17	11, 16	eleq12d 2695	. . . . . . . 8 ⊢ (𝑘 = 1𝑜 → ((𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺‘1𝑜) ∈ 𝐵))
18	5, 7, 10, 17	ralpr 4238	. . . . . . 7 ⊢ (∀𝑘 ∈ {∅, 1𝑜} (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵))
19	4, 18	bitri 264	. . . . . 6 ⊢ (∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵))
20		2onn 7720	. . . . . . . . . 10 ⊢ 2𝑜 ∈ ω
21		nnfi 8153	. . . . . . . . . 10 ⊢ (2𝑜 ∈ ω → 2𝑜 ∈ Fin)
22	20, 21	ax-mp 5	. . . . . . . . 9 ⊢ 2𝑜 ∈ Fin
23		fnfi 8238	. . . . . . . . 9 ⊢ ((𝐺 Fn 2𝑜 ∧ 2𝑜 ∈ Fin) → 𝐺 ∈ Fin)
24	22, 23	mpan2 707	. . . . . . . 8 ⊢ (𝐺 Fn 2𝑜 → 𝐺 ∈ Fin)
25		elex 3212	. . . . . . . 8 ⊢ (𝐺 ∈ Fin → 𝐺 ∈ V)
26	24, 25	syl 17	. . . . . . 7 ⊢ (𝐺 Fn 2𝑜 → 𝐺 ∈ V)
27	26	biantrurd 529	. . . . . 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 275	. . . . 5 ⊢ (𝐺 Fn 2𝑜 → ((𝐺 ∈ V ∧ ∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵)))
29	28	pm5.32i 669	. . . 4 ⊢ ((𝐺 Fn 2𝑜 ∧ (𝐺 ∈ V ∧ ∀𝑘 ∈ 2𝑜 (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))) ↔ (𝐺 Fn 2𝑜 ∧ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵)))
30		3anass 1042	. . . 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 1042	. . . 4 ⊢ ((𝐺 Fn 2𝑜 ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵) ↔ (𝐺 Fn 2𝑜 ∧ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵)))
32	29, 30, 31	3bitr4i 292	. . 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 264	. 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 264	1 ⊢ (𝐺 ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2𝑜 ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200  ∅c0 3915  ifcif 4086  {cpr 4179  Oncon0 5723   Fn wfn 5883  ‘cfv 5888  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554  Xcixp 7908  Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  xpsfrnel2  16225  xpsff1o  16228