Theorem fsnex 7291

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 7139	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑓:{𝐴}⟶𝐷 ↔ ((𝑓‘𝐴) ∈ 𝐷 ∧ 𝑓 = {⟨𝐴, (𝑓‘𝐴)⟩})))
2	1	simprbda 504	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓:{𝐴}⟶𝐷) → (𝑓‘𝐴) ∈ 𝐷)
3	2	adantrr 730	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → (𝑓‘𝐴) ∈ 𝐷)
4		fsnex.1	. . . . . . 7 ⊢ (𝑥 = (𝑓‘𝐴) → (𝜓 ↔ 𝜑))
5	4	adantl 487	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) ∧ 𝑥 = (𝑓‘𝐴)) → (𝜓 ↔ 𝜑))
6		simprr 785	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → 𝜑)
7	3, 5, 6	rspcedvd 3579	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → ∃𝑥 ∈ 𝐷 𝜓)
8	7	ex 418	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑓:{𝐴}⟶𝐷 ∧ 𝜑) → ∃𝑥 ∈ 𝐷 𝜓))
9	8	exlimdv 1966	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑) → ∃𝑥 ∈ 𝐷 𝜓))
10	9	imp 412	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑)) → ∃𝑥 ∈ 𝐷 𝜓)
11		nfv 1947	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
12		nfre1 3288	. . . 4 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐷 𝜓
13	11, 12	nfan 1932	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓)
14		f1osng 6867	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
15	14	elvd 3457	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
16	15	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
17		f1of 6824	. . . . . . 7 ⊢ ({⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥} → {⟨𝐴, 𝑥⟩}:{𝐴}⟶{𝑥})
18	16, 17	syl 18	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}⟶{𝑥})
19		simplr 781	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → 𝑥 ∈ 𝐷)
20	19	snssd 4747	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {𝑥} ⊆ 𝐷)
21	18, 20	fssd 6727	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷)
22		fvsng 7185	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
23	22	elvd 3457	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
24	23	eqcomd 2767	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))
25	24	ad3antrrr 743	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))
26		snex 5397	. . . . . 6 ⊢ {⟨𝐴, 𝑥⟩} ∈ V
27		feq1 6687	. . . . . . 7 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑓:{𝐴}⟶𝐷 ↔ {⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷))
28		fveq1 6884	. . . . . . . 8 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑓‘𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
29	28	eqeq2d 2772	. . . . . . 7 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑥 = (𝑓‘𝐴) ↔ 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴)))
30	27, 29	anbi12d 644	. . . . . 6 ⊢ (𝑓 = {⟨𝐴, 𝑥⟩} → ((𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)) ↔ ({⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷 ∧ 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))))
31	26, 30	spcev 3561	. . . . 5 ⊢ (({⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷 ∧ 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴)) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)))
32	21, 25, 31	syl2anc 596	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)))
33		simprl 783	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) → 𝑓:{𝐴}⟶𝐷)
34		simpllr 788	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → 𝜓)
35		simplrr 790	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → (𝜓 ↔ 𝜑))
37	34, 36	mpbid 235	. . . . . . . 8 ⊢ ((((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → 𝜑)
38	33, 37	mpdan 700	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) → 𝜑)
39	33, 38	jca 521	. . . . . 6 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴))) → (𝑓:{𝐴}⟶𝐷 ∧ 𝜑))
40	39	ex 418	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → ((𝑓:{𝐴}⟶𝐷 ∧ 𝑥 = (𝑓‘𝐴)) → (𝑓:{𝐴}⟶𝐷 ∧ 𝜑)))
41	40	eximdv 1950	. . . 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 16	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) ∧ 𝑥 ∈ 𝐷) ∧ 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑))
43		simpr 490	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) → ∃𝑥 ∈ 𝐷 𝜓)
44	13, 42, 43	r19.29af 3272	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑))
45	10, 44	impbida 813	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐷 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3087  Vcvv 3451  {csn 4584  ⟨cop 4590  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is used by: (None)