Theorem fsnex 7227

Description: Relate a function with a singleton as domain and one variable. (Contributed by Thierry Arnoux, 12-Jul-2020.)

Hypothesis

Ref	Expression
fsnex.1	⊢ (𝑥 = (𝑓‘𝐴) → (𝜓 ↔ 𝜑))

Assertion

Ref	Expression
fsnex	⊢ (𝐴 ∈ 𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐷 𝜓))

Distinct variable groups:   𝐴,𝑓,𝑥   𝐷,𝑓,𝑥   𝑓,𝑉,𝑥   𝜓,𝑓   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑥)

Proof of Theorem fsnex

Step	Hyp	Ref	Expression
1		fsn2g 7081	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑓:{𝐴}⟶𝐷 ↔ ((𝑓‘𝐴) ∈ 𝐷 ∧ 𝑓 = {⟨𝐴, (𝑓‘𝐴)⟩})))
2	1	simprbda 498	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓:{𝐴}⟶𝐷) → (𝑓‘𝐴) ∈ 𝐷)
3	2	adantrr 717	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → (𝑓‘𝐴) ∈ 𝐷)
4		fsnex.1	. . . . . . 7 ⊢ (𝑥 = (𝑓‘𝐴) → (𝜓 ↔ 𝜑))
5	4	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) ∧ 𝑥 = (𝑓‘𝐴)) → (𝜓 ↔ 𝜑))
6		simprr 772	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → 𝜑)
7	3, 5, 6	rspcedvd 3576	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → ∃𝑥 ∈ 𝐷 𝜓)
8	7	ex 412	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑓:{𝐴}⟶𝐷 ∧ 𝜑) → ∃𝑥 ∈ 𝐷 𝜓))
9	8	exlimdv 1934	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑) → ∃𝑥 ∈ 𝐷 𝜓))
10	9	imp 406	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → ∃𝑥 ∈ 𝐷 𝜓)
11		nfv 1915	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
12		nfre1 3259	. . . 4 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐷 𝜓
13	11, 12	nfan 1900	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓)
14		f1osng 6814	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
15	14	elvd 3444	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
16	15	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
17		f1of 6772	. . . . . . 7 ⊢ ({⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥} → {⟨𝐴, 𝑥⟩}:{𝐴}⟶{𝑥})
18	16, 17	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}⟶{𝑥})
19		simplr 768	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → 𝑥 ∈ 𝐷)
20	19	snssd 4763	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {𝑥} ⊆ 𝐷)
21	18, 20	fssd 6677	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷)
22		fvsng 7124	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
23	22	elvd 3444	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
24	23	eqcomd 2740	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))
25	24	ad3antrrr 730	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))
26		snex 5379	. . . . . 6 ⊢ {⟨𝐴, 𝑥⟩} ∈ V
27		feq1 6638	. . . . . . 7 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑓:{𝐴}⟶𝐷 ↔ {⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷))
28		fveq1 6831	. . . . . . . 8 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑓‘𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
29	28	eqeq2d 2745	. . . . . . 7 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑥 = (𝑓‘𝐴) ↔ 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴)))
30	27, 29	anbi12d 632	. . . . . 6 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → ((𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)) ↔ ({⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷 ∧ 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))))
31	26, 30	spcev 3558	. . . . 5 ⊢ (({⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷 ∧ 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴)) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)))
32	21, 25, 31	syl2anc 584	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)))
33		simprl 770	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) → 𝑓:{𝐴}⟶𝐷)
34		simpllr 775	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → 𝜓)
35		simplrr 777	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → 𝑥 = (𝑓‘𝐴))
36	35, 4	syl 17	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → (𝜓 ↔ 𝜑))
37	34, 36	mpbid 232	. . . . . . . 8 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → 𝜑)
38	33, 37	mpdan 687	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) → 𝜑)
39	33, 38	jca 511	. . . . . 6 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) → (𝑓:{𝐴}⟶𝐷 ∧ 𝜑))
40	39	ex 412	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → ((𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)) → (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)))
41	40	eximdv 1918	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → (∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑)))
42	32, 41	mpd 15	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑))
43		simpr 484	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) → ∃𝑥 ∈ 𝐷 𝜓)
44	13, 42, 43	r19.29af 3243	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑))
45	10, 44	impbida 800	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐷 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3058  Vcvv 3438  {csn 4578  ⟨cop 4584  ⟶wf 6486  –1-1-onto→wf1o 6489  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498

This theorem is referenced by: (None)