Theorem xpsfrnel2 16447

Description: Elementhood in the target space of the function 𝐹 appearing in xpsval 16454. (Contributed by Mario Carneiro, 15-Aug-2015.)

Assertion

Ref	Expression
xpsfrnel2	⊢ (◡({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑘,𝑋   𝑘,𝑌

Proof of Theorem xpsfrnel2

Step	Hyp	Ref	Expression
1		xpsfrnel 16445	. 2 ⊢ (◡({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵))
2		0ex 4942	. . . . . . . . . 10 ⊢ ∅ ∈ V
3	2	prid1 4441	. . . . . . . . 9 ⊢ ∅ ∈ {∅, 1𝑜}
4		df2o3 7744	. . . . . . . . 9 ⊢ 2𝑜 = {∅, 1𝑜}
5	3, 4	eleqtrri 2838	. . . . . . . 8 ⊢ ∅ ∈ 2𝑜
6		fndm 6151	. . . . . . . 8 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → dom ◡({𝑋} +𝑐 {𝑌}) = 2𝑜)
7	5, 6	syl5eleqr 2846	. . . . . . 7 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → ∅ ∈ dom ◡({𝑋} +𝑐 {𝑌}))
8		xpsc 16439	. . . . . . . . 9 ⊢ ◡({𝑋} +𝑐 {𝑌}) = (({∅} × {𝑋}) ∪ ({1𝑜} × {𝑌}))
9	8	dmeqi 5480	. . . . . . . 8 ⊢ dom ◡({𝑋} +𝑐 {𝑌}) = dom (({∅} × {𝑋}) ∪ ({1𝑜} × {𝑌}))
10		dmun 5486	. . . . . . . 8 ⊢ dom (({∅} × {𝑋}) ∪ ({1𝑜} × {𝑌})) = (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌}))
11	9, 10	eqtri 2782	. . . . . . 7 ⊢ dom ◡({𝑋} +𝑐 {𝑌}) = (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌}))
12	7, 11	syl6eleq 2849	. . . . . 6 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → ∅ ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})))
13		elun 3896	. . . . . . 7 ⊢ (∅ ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) ↔ (∅ ∈ dom ({∅} × {𝑋}) ∨ ∅ ∈ dom ({1𝑜} × {𝑌})))
14	2	eldm 5476	. . . . . . . . 9 ⊢ (∅ ∈ dom ({∅} × {𝑋}) ↔ ∃𝑘∅({∅} × {𝑋})𝑘)
15		brxp 5304	. . . . . . . . . . 11 ⊢ (∅({∅} × {𝑋})𝑘 ↔ (∅ ∈ {∅} ∧ 𝑘 ∈ {𝑋}))
16		elsni 4338	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝑋} → 𝑘 = 𝑋)
17		vex 3343	. . . . . . . . . . . . 13 ⊢ 𝑘 ∈ V
18	16, 17	syl6eqelr 2848	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑋} → 𝑋 ∈ V)
19	18	adantl 473	. . . . . . . . . . 11 ⊢ ((∅ ∈ {∅} ∧ 𝑘 ∈ {𝑋}) → 𝑋 ∈ V)
20	15, 19	sylbi 207	. . . . . . . . . 10 ⊢ (∅({∅} × {𝑋})𝑘 → 𝑋 ∈ V)
21	20	exlimiv 2007	. . . . . . . . 9 ⊢ (∃𝑘∅({∅} × {𝑋})𝑘 → 𝑋 ∈ V)
22	14, 21	sylbi 207	. . . . . . . 8 ⊢ (∅ ∈ dom ({∅} × {𝑋}) → 𝑋 ∈ V)
23		dmxpss 5723	. . . . . . . . . 10 ⊢ dom ({1𝑜} × {𝑌}) ⊆ {1𝑜}
24	23	sseli 3740	. . . . . . . . 9 ⊢ (∅ ∈ dom ({1𝑜} × {𝑌}) → ∅ ∈ {1𝑜})
25		elsni 4338	. . . . . . . . 9 ⊢ (∅ ∈ {1𝑜} → ∅ = 1𝑜)
26		1n0 7746	. . . . . . . . . . . 12 ⊢ 1𝑜 ≠ ∅
27	26	neii 2934	. . . . . . . . . . 11 ⊢ ¬ 1𝑜 = ∅
28	27	pm2.21i 116	. . . . . . . . . 10 ⊢ (1𝑜 = ∅ → 𝑋 ∈ V)
29	28	eqcoms 2768	. . . . . . . . 9 ⊢ (∅ = 1𝑜 → 𝑋 ∈ V)
30	24, 25, 29	3syl 18	. . . . . . . 8 ⊢ (∅ ∈ dom ({1𝑜} × {𝑌}) → 𝑋 ∈ V)
31	22, 30	jaoi 393	. . . . . . 7 ⊢ ((∅ ∈ dom ({∅} × {𝑋}) ∨ ∅ ∈ dom ({1𝑜} × {𝑌})) → 𝑋 ∈ V)
32	13, 31	sylbi 207	. . . . . 6 ⊢ (∅ ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) → 𝑋 ∈ V)
33	12, 32	syl 17	. . . . 5 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 𝑋 ∈ V)
34		1oex 7738	. . . . . . . . . 10 ⊢ 1𝑜 ∈ V
35	34	prid2 4442	. . . . . . . . 9 ⊢ 1𝑜 ∈ {∅, 1𝑜}
36	35, 4	eleqtrri 2838	. . . . . . . 8 ⊢ 1𝑜 ∈ 2𝑜
37	36, 6	syl5eleqr 2846	. . . . . . 7 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 1𝑜 ∈ dom ◡({𝑋} +𝑐 {𝑌}))
38	37, 11	syl6eleq 2849	. . . . . 6 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 1𝑜 ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})))
39		elun 3896	. . . . . . 7 ⊢ (1𝑜 ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) ↔ (1𝑜 ∈ dom ({∅} × {𝑋}) ∨ 1𝑜 ∈ dom ({1𝑜} × {𝑌})))
40		dmxpss 5723	. . . . . . . . . 10 ⊢ dom ({∅} × {𝑋}) ⊆ {∅}
41	40	sseli 3740	. . . . . . . . 9 ⊢ (1𝑜 ∈ dom ({∅} × {𝑋}) → 1𝑜 ∈ {∅})
42		elsni 4338	. . . . . . . . 9 ⊢ (1𝑜 ∈ {∅} → 1𝑜 = ∅)
43	27	pm2.21i 116	. . . . . . . . 9 ⊢ (1𝑜 = ∅ → 𝑌 ∈ V)
44	41, 42, 43	3syl 18	. . . . . . . 8 ⊢ (1𝑜 ∈ dom ({∅} × {𝑋}) → 𝑌 ∈ V)
45	34	eldm 5476	. . . . . . . . 9 ⊢ (1𝑜 ∈ dom ({1𝑜} × {𝑌}) ↔ ∃𝑘1𝑜({1𝑜} × {𝑌})𝑘)
46		brxp 5304	. . . . . . . . . . 11 ⊢ (1𝑜({1𝑜} × {𝑌})𝑘 ↔ (1𝑜 ∈ {1𝑜} ∧ 𝑘 ∈ {𝑌}))
47		elsni 4338	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝑌} → 𝑘 = 𝑌)
48	47, 17	syl6eqelr 2848	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑌} → 𝑌 ∈ V)
49	48	adantl 473	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ {1𝑜} ∧ 𝑘 ∈ {𝑌}) → 𝑌 ∈ V)
50	46, 49	sylbi 207	. . . . . . . . . 10 ⊢ (1𝑜({1𝑜} × {𝑌})𝑘 → 𝑌 ∈ V)
51	50	exlimiv 2007	. . . . . . . . 9 ⊢ (∃𝑘1𝑜({1𝑜} × {𝑌})𝑘 → 𝑌 ∈ V)
52	45, 51	sylbi 207	. . . . . . . 8 ⊢ (1𝑜 ∈ dom ({1𝑜} × {𝑌}) → 𝑌 ∈ V)
53	44, 52	jaoi 393	. . . . . . 7 ⊢ ((1𝑜 ∈ dom ({∅} × {𝑋}) ∨ 1𝑜 ∈ dom ({1𝑜} × {𝑌})) → 𝑌 ∈ V)
54	39, 53	sylbi 207	. . . . . 6 ⊢ (1𝑜 ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) → 𝑌 ∈ V)
55	38, 54	syl 17	. . . . 5 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 𝑌 ∈ V)
56	33, 55	jca 555	. . . 4 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
57	56	3ad2ant1 1128	. . 3 ⊢ ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
58		elex 3352	. . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
59		elex 3352	. . . 4 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ V)
60	58, 59	anim12i 591	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
61		3anass 1081	. . . 4 ⊢ ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ ((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵)))
62		xpscfn 16441	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜)
63	62	biantrurd 530	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ ((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵))))
64		xpsc0 16442	. . . . . . 7 ⊢ (𝑋 ∈ V → (◡({𝑋} +𝑐 {𝑌})‘∅) = 𝑋)
65	64	eleq1d 2824	. . . . . 6 ⊢ (𝑋 ∈ V → ((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
66		xpsc1 16443	. . . . . . 7 ⊢ (𝑌 ∈ V → (◡({𝑋} +𝑐 {𝑌})‘1𝑜) = 𝑌)
67	66	eleq1d 2824	. . . . . 6 ⊢ (𝑌 ∈ V → ((◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵 ↔ 𝑌 ∈ 𝐵))
68	65, 67	bi2anan9 953	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)))
69	63, 68	bitr3d 270	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ ((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵)) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)))
70	61, 69	syl5bb 272	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)))
71	57, 60, 70	pm5.21nii 367	. 2 ⊢ ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
72	1, 71	bitri 264	1 ⊢ (◡({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))

Syntax hints:   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1632  ∃wex 1853   ∈ wcel 2139  Vcvv 3340   ∪ cun 3713  ∅c0 4058  ifcif 4230  {csn 4321  {cpr 4323   class class class wbr 4804   × cxp 5264  ^◡ccnv 5265  dom cdm 5266   Fn wfn 6044  ‘cfv 6049  (class class class)co 6814  1_𝑜c1o 7723  2_𝑜c2o 7724  Xcixp 8076   +_𝑐 ccda 9201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-ixp 8077  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-cda 9202

This theorem is referenced by:  xpscf  16448  xpsff1o  16450