Theorem cfsetsnfsetf1 44440

Description: The mapping of the class of singleton functions into the class of constant functions is an injection. (Contributed by AV, 14-Sep-2024.)

Hypotheses

Ref	Expression
cfsetsnfsetfv.f	⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}
cfsetsnfsetfv.g	⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}
cfsetsnfsetfv.h	⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))

Assertion

Ref	Expression
cfsetsnfsetf1	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1→𝐹)

Distinct variable groups:   𝐴,𝑎,𝑔   𝑔,𝐺   𝑔,𝑉   𝑔,𝑌   𝐴,𝑏,𝑓,𝑧   𝑥,𝐵   𝐵,𝑎,𝑏,𝑓   𝑔,𝐹   𝐺,𝑎,𝑏,𝑧   𝑉,𝑎,𝑏,𝑧   𝑌,𝑎,𝑏,𝑓,𝑧   𝑥,𝑌,𝑔   𝑔,𝑏,𝑓,𝑧

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑧,𝑔)   𝐹(𝑥,𝑧,𝑓,𝑎,𝑏)   𝐺(𝑥,𝑓)   𝐻(𝑥,𝑧,𝑓,𝑔,𝑎,𝑏)   𝑉(𝑥,𝑓)

Proof of Theorem cfsetsnfsetf1

Dummy variables 𝑚 𝑛 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfsetsnfsetfv.f	. . 3 ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}
2		cfsetsnfsetfv.g	. . 3 ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}
3		cfsetsnfsetfv.h	. . 3 ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))
4	1, 2, 3	cfsetsnfsetf 44439	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)
5	1, 2, 3	cfsetsnfsetfv 44438	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑚 ∈ 𝐺) → (𝐻‘𝑚) = (𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)))
6	5	ad2ant2r 743	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → (𝐻‘𝑚) = (𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)))
7	1, 2, 3	cfsetsnfsetfv 44438	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ 𝐺) → (𝐻‘𝑛) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
8	7	ad2ant2rl 745	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → (𝐻‘𝑛) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
9	6, 8	eqeq12d 2754	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝐻‘𝑚) = (𝐻‘𝑛) ↔ (𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌))))
10		fvexd 6771	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) ∧ 𝑎 ∈ 𝐴) → (𝑚‘𝑌) ∈ V)
11	10	ralrimiva 3107	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ∀𝑎 ∈ 𝐴 (𝑚‘𝑌) ∈ V)
12		mpteqb 6876	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 (𝑚‘𝑌) ∈ V → ((𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) ↔ ∀𝑎 ∈ 𝐴 (𝑚‘𝑌) = (𝑛‘𝑌)))
13	11, 12	syl 17	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) ↔ ∀𝑎 ∈ 𝐴 (𝑚‘𝑌) = (𝑛‘𝑌)))
14		simplr 765	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → 𝑌 ∈ 𝐴)
15		idd 24	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) ∧ 𝑎 = 𝑌) → ((𝑚‘𝑌) = (𝑛‘𝑌) → (𝑚‘𝑌) = (𝑛‘𝑌)))
16	14, 15	rspcimdv 3541	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → (∀𝑎 ∈ 𝐴 (𝑚‘𝑌) = (𝑛‘𝑌) → (𝑚‘𝑌) = (𝑛‘𝑌)))
17		vex 3426	. . . . . . . . . 10 ⊢ 𝑚 ∈ V
18		feq1 6565	. . . . . . . . . 10 ⊢ (𝑥 = 𝑚 → (𝑥:{𝑌}⟶𝐵 ↔ 𝑚:{𝑌}⟶𝐵))
19	17, 18, 2	elab2 3606	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝐺 ↔ 𝑚:{𝑌}⟶𝐵)
20		vex 3426	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
21		feq1 6565	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝑥:{𝑌}⟶𝐵 ↔ 𝑛:{𝑌}⟶𝐵))
22	20, 21, 2	elab2 3606	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐺 ↔ 𝑛:{𝑌}⟶𝐵)
23	19, 22	anbi12i 626	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺) ↔ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵))
24		simp3 1136	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → (𝑚‘𝑌) = (𝑛‘𝑌))
25		simp1r 1196	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → 𝑌 ∈ 𝐴)
26		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑚‘𝑦) = (𝑚‘𝑌))
27		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑛‘𝑦) = (𝑛‘𝑌))
28	26, 27	eqeq12d 2754	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → ((𝑚‘𝑦) = (𝑛‘𝑦) ↔ (𝑚‘𝑌) = (𝑛‘𝑌)))
29	28	ralsng 4606	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ 𝐴 → (∀𝑦 ∈ {𝑌} (𝑚‘𝑦) = (𝑛‘𝑦) ↔ (𝑚‘𝑌) = (𝑛‘𝑌)))
30	25, 29	syl 17	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → (∀𝑦 ∈ {𝑌} (𝑚‘𝑦) = (𝑛‘𝑦) ↔ (𝑚‘𝑌) = (𝑛‘𝑌)))
31	24, 30	mpbird 256	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → ∀𝑦 ∈ {𝑌} (𝑚‘𝑦) = (𝑛‘𝑦))
32		ffn 6584	. . . . . . . . . . . . 13 ⊢ (𝑚:{𝑌}⟶𝐵 → 𝑚 Fn {𝑌})
33		ffn 6584	. . . . . . . . . . . . 13 ⊢ (𝑛:{𝑌}⟶𝐵 → 𝑛 Fn {𝑌})
34	32, 33	anim12i 612	. . . . . . . . . . . 12 ⊢ ((𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) → (𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}))
35	34	3ad2ant2 1132	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → (𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}))
36		eqfnfv 6891	. . . . . . . . . . 11 ⊢ ((𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}) → (𝑚 = 𝑛 ↔ ∀𝑦 ∈ {𝑌} (𝑚‘𝑦) = (𝑛‘𝑦)))
37	35, 36	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → (𝑚 = 𝑛 ↔ ∀𝑦 ∈ {𝑌} (𝑚‘𝑦) = (𝑛‘𝑦)))
38	31, 37	mpbird 256	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → 𝑚 = 𝑛)
39	38	3exp 1117	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) → ((𝑚‘𝑌) = (𝑛‘𝑌) → 𝑚 = 𝑛)))
40	23, 39	syl5bi 241	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺) → ((𝑚‘𝑌) = (𝑛‘𝑌) → 𝑚 = 𝑛)))
41	40	imp 406	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝑚‘𝑌) = (𝑛‘𝑌) → 𝑚 = 𝑛))
42	16, 41	syld 47	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → (∀𝑎 ∈ 𝐴 (𝑚‘𝑌) = (𝑛‘𝑌) → 𝑚 = 𝑛))
43	13, 42	sylbid 239	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) → 𝑚 = 𝑛))
44	9, 43	sylbid 239	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝐻‘𝑚) = (𝐻‘𝑛) → 𝑚 = 𝑛))
45	44	ralrimivva 3114	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ∀𝑚 ∈ 𝐺 ∀𝑛 ∈ 𝐺 ((𝐻‘𝑚) = (𝐻‘𝑛) → 𝑚 = 𝑛))
46		dff13 7109	. 2 ⊢ (𝐻:𝐺–1-1→𝐹 ↔ (𝐻:𝐺⟶𝐹 ∧ ∀𝑚 ∈ 𝐺 ∀𝑛 ∈ 𝐺 ((𝐻‘𝑚) = (𝐻‘𝑛) → 𝑚 = 𝑛)))
47	4, 45, 46	sylanbrc 582	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1→𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  Vcvv 3422  {csn 4558   ↦ cmpt 5153   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426

This theorem is referenced by:  cfsetsnfsetf1o  44442