Theorem elixpsn 6909

Description: Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Assertion

Ref	Expression
elixpsn	⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦   𝑥,𝑉,𝑦

Proof of Theorem elixpsn

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3681	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑧} = {𝐴})
2	1	ixpeq1d 6884	. . 3 ⊢ (𝑧 = 𝐴 → X𝑥 ∈ {𝑧}𝐵 = X𝑥 ∈ {𝐴}𝐵)
3	2	eleq2d 2300	. 2 ⊢ (𝑧 = 𝐴 → (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ 𝐹 ∈ X𝑥 ∈ {𝐴}𝐵))
4		opeq1 3863	. . . . 5 ⊢ (𝑧 = 𝐴 → ⟨𝑧, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
5	4	sneqd 3683	. . . 4 ⊢ (𝑧 = 𝐴 → {⟨𝑧, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
6	5	eqeq2d 2242	. . 3 ⊢ (𝑧 = 𝐴 → (𝐹 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, 𝑦⟩}))
7	6	rexbidv 2532	. 2 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
8		elex 2813	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 → 𝐹 ∈ V)
9		vex 2804	. . . . . . 7 ⊢ 𝑧 ∈ V
10		vex 2804	. . . . . . 7 ⊢ 𝑦 ∈ V
11	9, 10	opex 4323	. . . . . 6 ⊢ ⟨𝑧, 𝑦⟩ ∈ V
12	11	snex 4277	. . . . 5 ⊢ {⟨𝑧, 𝑦⟩} ∈ V
13		eleq1 2293	. . . . 5 ⊢ (𝐹 = {⟨𝑧, 𝑦⟩} → (𝐹 ∈ V ↔ {⟨𝑧, 𝑦⟩} ∈ V))
14	12, 13	mpbiri 168	. . . 4 ⊢ (𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
15	14	rexlimivw 2645	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
16		eleq1 2293	. . . 4 ⊢ (𝑤 = 𝐹 → (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ 𝐹 ∈ X𝑥 ∈ {𝑧}𝐵))
17		eqeq1 2237	. . . . 5 ⊢ (𝑤 = 𝐹 → (𝑤 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝑧, 𝑦⟩}))
18	17	rexbidv 2532	. . . 4 ⊢ (𝑤 = 𝐹 → (∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
19		vex 2804	. . . . . 6 ⊢ 𝑤 ∈ V
20	19	elixp 6879	. . . . 5 ⊢ (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵))
21		fveq2 5642	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑤‘𝑥) = (𝑤‘𝑧))
22	21	eleq1d 2299	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑤‘𝑥) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵))
23	9, 22	ralsn 3713	. . . . . 6 ⊢ (∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵)
24	23	anbi2i 457	. . . . 5 ⊢ ((𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵) ↔ (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵))
25		simpl 109	. . . . . . . . 9 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → 𝑤 Fn {𝑧})
26		fveq2 5642	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑤‘𝑦) = (𝑤‘𝑧))
27	26	eleq1d 2299	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ((𝑤‘𝑦) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵))
28	9, 27	ralsn 3713	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵)
29	28	biimpri 133	. . . . . . . . . 10 ⊢ ((𝑤‘𝑧) ∈ 𝐵 → ∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵)
30	29	adantl 277	. . . . . . . . 9 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵)
31		ffnfv 5808	. . . . . . . . 9 ⊢ (𝑤:{𝑧}⟶𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵))
32	25, 30, 31	sylanbrc 417	. . . . . . . 8 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → 𝑤:{𝑧}⟶𝐵)
33	9	fsn2 5824	. . . . . . . 8 ⊢ (𝑤:{𝑧}⟶𝐵 ↔ ((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}))
34	32, 33	sylib 122	. . . . . . 7 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}))
35		opeq2 3864	. . . . . . . . 9 ⊢ (𝑦 = (𝑤‘𝑧) → ⟨𝑧, 𝑦⟩ = ⟨𝑧, (𝑤‘𝑧)⟩)
36	35	sneqd 3683	. . . . . . . 8 ⊢ (𝑦 = (𝑤‘𝑧) → {⟨𝑧, 𝑦⟩} = {⟨𝑧, (𝑤‘𝑧)⟩})
37	36	rspceeqv 2927	. . . . . . 7 ⊢ (((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}) → ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
38	34, 37	syl 14	. . . . . 6 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
39	9, 10	fvsn 5852	. . . . . . . . . 10 ⊢ ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
40		id 19	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐵)
41	39, 40	eqeltrid 2317	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)
42	9, 10	fnsn 5386	. . . . . . . . 9 ⊢ {⟨𝑧, 𝑦⟩} Fn {𝑧}
43	41, 42	jctil 312	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
44		fneq1 5420	. . . . . . . . 9 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ↔ {⟨𝑧, 𝑦⟩} Fn {𝑧}))
45		fveq1 5641	. . . . . . . . . 10 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
46	45	eleq1d 2299	. . . . . . . . 9 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤‘𝑧) ∈ 𝐵 ↔ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
47	44, 46	anbi12d 473	. . . . . . . 8 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) ↔ ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)))
48	43, 47	syl5ibrcom 157	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵)))
49	48	rexlimiv 2643	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵))
50	38, 49	impbii 126	. . . . 5 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
51	20, 24, 50	3bitri 206	. . . 4 ⊢ (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
52	16, 18, 51	vtoclbg 2864	. . 3 ⊢ (𝐹 ∈ V → (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
53	8, 15, 52	pm5.21nii 711	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩})
54	3, 7, 53	vtoclbg 2864	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2201  ∀wral 2509  ∃wrex 2510  Vcvv 2801  {csn 3670  ⟨cop 3673   Fn wfn 5323  ⟶wf 5324  ‘cfv 5328  Xcixp 6872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-reu 2516  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ixp 6873

This theorem is referenced by:  ixpsnf1o  6910