Theorem xpsfrnel2 16048

Description: Elementhood in the target space of the function 𝐹 appearing in xpsval 16055. (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 16046	. 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 4718	. . . . . . . . . 10 ⊢ ∅ ∈ V
3	2	prid1 4241	. . . . . . . . 9 ⊢ ∅ ∈ {∅, 1𝑜}
4		df2o3 7460	. . . . . . . . 9 ⊢ 2𝑜 = {∅, 1𝑜}
5	3, 4	eleqtrri 2687	. . . . . . . 8 ⊢ ∅ ∈ 2𝑜
6		fndm 5904	. . . . . . . 8 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → dom ◡({𝑋} +𝑐 {𝑌}) = 2𝑜)
7	5, 6	syl5eleqr 2695	. . . . . . 7 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → ∅ ∈ dom ◡({𝑋} +𝑐 {𝑌}))
8		xpsc 16040	. . . . . . . . 9 ⊢ ◡({𝑋} +𝑐 {𝑌}) = (({∅} × {𝑋}) ∪ ({1𝑜} × {𝑌}))
9	8	dmeqi 5247	. . . . . . . 8 ⊢ dom ◡({𝑋} +𝑐 {𝑌}) = dom (({∅} × {𝑋}) ∪ ({1𝑜} × {𝑌}))
10		dmun 5253	. . . . . . . 8 ⊢ dom (({∅} × {𝑋}) ∪ ({1𝑜} × {𝑌})) = (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌}))
11	9, 10	eqtri 2632	. . . . . . 7 ⊢ dom ◡({𝑋} +𝑐 {𝑌}) = (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌}))
12	7, 11	syl6eleq 2698	. . . . . 6 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → ∅ ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})))
13		elun 3715	. . . . . . 7 ⊢ (∅ ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) ↔ (∅ ∈ dom ({∅} × {𝑋}) ∨ ∅ ∈ dom ({1𝑜} × {𝑌})))
14	2	eldm 5243	. . . . . . . . 9 ⊢ (∅ ∈ dom ({∅} × {𝑋}) ↔ ∃𝑘∅({∅} × {𝑋})𝑘)
15		brxp 5071	. . . . . . . . . . 11 ⊢ (∅({∅} × {𝑋})𝑘 ↔ (∅ ∈ {∅} ∧ 𝑘 ∈ {𝑋}))
16		elsni 4142	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝑋} → 𝑘 = 𝑋)
17		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑘 ∈ V
18	16, 17	syl6eqelr 2697	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑋} → 𝑋 ∈ V)
19	18	adantl 481	. . . . . . . . . . 11 ⊢ ((∅ ∈ {∅} ∧ 𝑘 ∈ {𝑋}) → 𝑋 ∈ V)
20	15, 19	sylbi 206	. . . . . . . . . 10 ⊢ (∅({∅} × {𝑋})𝑘 → 𝑋 ∈ V)
21	20	exlimiv 1845	. . . . . . . . 9 ⊢ (∃𝑘∅({∅} × {𝑋})𝑘 → 𝑋 ∈ V)
22	14, 21	sylbi 206	. . . . . . . 8 ⊢ (∅ ∈ dom ({∅} × {𝑋}) → 𝑋 ∈ V)
23		dmxpss 5484	. . . . . . . . . 10 ⊢ dom ({1𝑜} × {𝑌}) ⊆ {1𝑜}
24	23	sseli 3564	. . . . . . . . 9 ⊢ (∅ ∈ dom ({1𝑜} × {𝑌}) → ∅ ∈ {1𝑜})
25		elsni 4142	. . . . . . . . 9 ⊢ (∅ ∈ {1𝑜} → ∅ = 1𝑜)
26		1n0 7462	. . . . . . . . . . . 12 ⊢ 1𝑜 ≠ ∅
27	26	neii 2784	. . . . . . . . . . 11 ⊢ ¬ 1𝑜 = ∅
28	27	pm2.21i 115	. . . . . . . . . 10 ⊢ (1𝑜 = ∅ → 𝑋 ∈ V)
29	28	eqcoms 2618	. . . . . . . . 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 206	. . . . . 6 ⊢ (∅ ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) → 𝑋 ∈ V)
33	12, 32	syl 17	. . . . 5 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 𝑋 ∈ V)
34		1on 7454	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ On
35	34	elexi 3186	. . . . . . . . . 10 ⊢ 1𝑜 ∈ V
36	35	prid2 4242	. . . . . . . . 9 ⊢ 1𝑜 ∈ {∅, 1𝑜}
37	36, 4	eleqtrri 2687	. . . . . . . 8 ⊢ 1𝑜 ∈ 2𝑜
38	37, 6	syl5eleqr 2695	. . . . . . 7 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 1𝑜 ∈ dom ◡({𝑋} +𝑐 {𝑌}))
39	38, 11	syl6eleq 2698	. . . . . 6 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 1𝑜 ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})))
40		elun 3715	. . . . . . 7 ⊢ (1𝑜 ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) ↔ (1𝑜 ∈ dom ({∅} × {𝑋}) ∨ 1𝑜 ∈ dom ({1𝑜} × {𝑌})))
41		dmxpss 5484	. . . . . . . . . 10 ⊢ dom ({∅} × {𝑋}) ⊆ {∅}
42	41	sseli 3564	. . . . . . . . 9 ⊢ (1𝑜 ∈ dom ({∅} × {𝑋}) → 1𝑜 ∈ {∅})
43		elsni 4142	. . . . . . . . 9 ⊢ (1𝑜 ∈ {∅} → 1𝑜 = ∅)
44	27	pm2.21i 115	. . . . . . . . 9 ⊢ (1𝑜 = ∅ → 𝑌 ∈ V)
45	42, 43, 44	3syl 18	. . . . . . . 8 ⊢ (1𝑜 ∈ dom ({∅} × {𝑋}) → 𝑌 ∈ V)
46	35	eldm 5243	. . . . . . . . 9 ⊢ (1𝑜 ∈ dom ({1𝑜} × {𝑌}) ↔ ∃𝑘1𝑜({1𝑜} × {𝑌})𝑘)
47		brxp 5071	. . . . . . . . . . 11 ⊢ (1𝑜({1𝑜} × {𝑌})𝑘 ↔ (1𝑜 ∈ {1𝑜} ∧ 𝑘 ∈ {𝑌}))
48		elsni 4142	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝑌} → 𝑘 = 𝑌)
49	48, 17	syl6eqelr 2697	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑌} → 𝑌 ∈ V)
50	49	adantl 481	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ {1𝑜} ∧ 𝑘 ∈ {𝑌}) → 𝑌 ∈ V)
51	47, 50	sylbi 206	. . . . . . . . . 10 ⊢ (1𝑜({1𝑜} × {𝑌})𝑘 → 𝑌 ∈ V)
52	51	exlimiv 1845	. . . . . . . . 9 ⊢ (∃𝑘1𝑜({1𝑜} × {𝑌})𝑘 → 𝑌 ∈ V)
53	46, 52	sylbi 206	. . . . . . . 8 ⊢ (1𝑜 ∈ dom ({1𝑜} × {𝑌}) → 𝑌 ∈ V)
54	45, 53	jaoi 393	. . . . . . 7 ⊢ ((1𝑜 ∈ dom ({∅} × {𝑋}) ∨ 1𝑜 ∈ dom ({1𝑜} × {𝑌})) → 𝑌 ∈ V)
55	40, 54	sylbi 206	. . . . . 6 ⊢ (1𝑜 ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) → 𝑌 ∈ V)
56	39, 55	syl 17	. . . . 5 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 𝑌 ∈ V)
57	33, 56	jca 553	. . . 4 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
58	57	3ad2ant1 1075	. . 3 ⊢ ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
59		elex 3185	. . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
60		elex 3185	. . . 4 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ V)
61	59, 60	anim12i 588	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
62		3anass 1035	. . . 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)))
63		xpscfn 16042	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜)
64	63	biantrurd 528	. . . . 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))))
65		xpsc0 16043	. . . . . . 7 ⊢ (𝑋 ∈ V → (◡({𝑋} +𝑐 {𝑌})‘∅) = 𝑋)
66	65	eleq1d 2672	. . . . . 6 ⊢ (𝑋 ∈ V → ((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
67		xpsc1 16044	. . . . . . 7 ⊢ (𝑌 ∈ V → (◡({𝑋} +𝑐 {𝑌})‘1𝑜) = 𝑌)
68	67	eleq1d 2672	. . . . . 6 ⊢ (𝑌 ∈ V → ((◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵 ↔ 𝑌 ∈ 𝐵))
69	66, 68	bi2anan9 913	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)))
70	64, 69	bitr3d 269	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ ((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵)) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)))
71	62, 70	syl5bb 271	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)))
72	58, 61, 71	pm5.21nii 367	. 2 ⊢ ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
73	1, 72	bitri 263	1 ⊢ (◡({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))

Syntax hints:   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538  ∅c0 3874  ifcif 4036  {csn 4125  {cpr 4127   class class class wbr 4583   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  Oncon0 5640   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 7441  Xcixp 7794   +_𝑐 ccda 8872

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  df-cda 8873

This theorem is referenced by:  xpscf  16049  xpsff1o  16051