Theorem fsnex 6538

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 6405	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑓:{𝐴}⟶𝐷 ↔ ((𝑓‘𝐴) ∈ 𝐷 ∧ 𝑓 = {⟨𝐴, (𝑓‘𝐴)⟩})))
2	1	simprbda 653	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓:{𝐴}⟶𝐷) → (𝑓‘𝐴) ∈ 𝐷)
3	2	adantrr 753	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → (𝑓‘𝐴) ∈ 𝐷)
4		fsnex.1	. . . . . . 7 ⊢ (𝑥 = (𝑓‘𝐴) → (𝜓 ↔ 𝜑))
5	4	adantl 482	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) ∧ 𝑥 = (𝑓‘𝐴)) → (𝜓 ↔ 𝜑))
6		simprr 796	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → 𝜑)
7	3, 5, 6	rspcedvd 3317	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → ∃𝑥 ∈ 𝐷 𝜓)
8	7	ex 450	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑓:{𝐴}⟶𝐷 ∧ 𝜑) → ∃𝑥 ∈ 𝐷 𝜓))
9	8	exlimdv 1861	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑) → ∃𝑥 ∈ 𝐷 𝜓))
10	9	imp 445	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → ∃𝑥 ∈ 𝐷 𝜓)
11		nfv 1843	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
12		nfre1 3005	. . . 4 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐷 𝜓
13	11, 12	nfan 1828	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓)
14		vex 3203	. . . . . . . . 9 ⊢ 𝑥 ∈ V
15		f1osng 6177	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
16	14, 15	mpan2 707	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
17	16	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
18		f1of 6137	. . . . . . 7 ⊢ ({⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥} → {⟨𝐴, 𝑥⟩}:{𝐴}⟶{𝑥})
19	17, 18	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}⟶{𝑥})
20		simplr 792	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → 𝑥 ∈ 𝐷)
21	20	snssd 4340	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {𝑥} ⊆ 𝐷)
22		fss 6056	. . . . . 6 ⊢ (({⟨𝐴, 𝑥⟩}:{𝐴}⟶{𝑥} ∧ {𝑥} ⊆ 𝐷) → {⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷)
23	19, 21, 22	syl2anc 693	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷)
24		fvsng 6447	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
25	14, 24	mpan2 707	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
26	25	eqcomd 2628	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))
27	26	ad3antrrr 766	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))
28		snex 4908	. . . . . 6 ⊢ {⟨𝐴, 𝑥⟩} ∈ V
29		feq1 6026	. . . . . . 7 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑓:{𝐴}⟶𝐷 ↔ {⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷))
30		fveq1 6190	. . . . . . . 8 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑓‘𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
31	30	eqeq2d 2632	. . . . . . 7 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑥 = (𝑓‘𝐴) ↔ 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴)))
32	29, 31	anbi12d 747	. . . . . 6 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → ((𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)) ↔ ({⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷 ∧ 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))))
33	28, 32	spcev 3300	. . . . 5 ⊢ (({⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷 ∧ 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴)) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)))
34	23, 27, 33	syl2anc 693	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)))
35		simprl 794	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) → 𝑓:{𝐴}⟶𝐷)
36		simpllr 799	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → 𝜓)
37		simplrr 801	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → 𝑥 = (𝑓‘𝐴))
38	37, 4	syl 17	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → (𝜓 ↔ 𝜑))
39	36, 38	mpbid 222	. . . . . . . 8 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → 𝜑)
40	35, 39	mpdan 702	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) → 𝜑)
41	35, 40	jca 554	. . . . . 6 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) → (𝑓:{𝐴}⟶𝐷 ∧ 𝜑))
42	41	ex 450	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → ((𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)) → (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)))
43	42	eximdv 1846	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → (∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑)))
44	34, 43	mpd 15	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑))
45		simpr 477	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) → ∃𝑥 ∈ 𝐷 𝜓)
46	13, 44, 45	r19.29af 3076	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑))
47	10, 46	impbida 877	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐷 𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  {csn 4177  ⟨cop 4183  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by: (None)