Theorem cfsetsnfsetf1 48223

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 48222	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)
5	1, 2, 3	cfsetsnfsetfv 48221	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑚 ∈ 𝐺) → (𝐻‘𝑚) = (𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)))
6	5	ad2ant2r 760	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → (𝐻‘𝑚) = (𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)))
7	1, 2, 3	cfsetsnfsetfv 48221	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ 𝐺) → (𝐻‘𝑛) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
8	7	ad2ant2rl 762	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → (𝐻‘𝑛) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
9	6, 8	eqeq12d 2776	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝐻‘𝑚) = (𝐻‘𝑛) ↔ (𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌))))
10		fvexd 6901	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) ∧ 𝑎 ∈ 𝐴) → (𝑚‘𝑌) ∈ V)
11	10	ralrimiva 3154	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ∀𝑎 ∈ 𝐴 (𝑚‘𝑌) ∈ V)
12		mpteqb 7014	. . . . . 6 ⊢ (∀𝑎 ∈ 𝐴 (𝑚‘𝑌) ∈ V → ((𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) ↔ ∀𝑎 ∈ 𝐴 (𝑚‘𝑌) = (𝑛‘𝑌)))
13	11, 12	syl 18	. . . . 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 781	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → 𝑌 ∈ 𝐴)
15		idd 25	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) ∧ 𝑎 = 𝑌) → ((𝑚‘𝑌) = (𝑛‘𝑌) → (𝑚‘𝑌) = (𝑛‘𝑌)))
16	14, 15	rspcimdv 3566	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → (∀𝑎 ∈ 𝐴 (𝑚‘𝑌) = (𝑛‘𝑌) → (𝑚‘𝑌) = (𝑛‘𝑌)))
17		vex 3454	. . . . . . . . . 10 ⊢ 𝑚 ∈ V
18		feq1 6688	. . . . . . . . . 10 ⊢ (𝑥 = 𝑚 → (𝑥:{𝑌}⟶𝐵 ↔ 𝑚:{𝑌}⟶𝐵))
19	17, 18, 2	elab2 3635	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝐺 ↔ 𝑚:{𝑌}⟶𝐵)
20		vex 3454	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
21		feq1 6688	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝑥:{𝑌}⟶𝐵 ↔ 𝑛:{𝑌}⟶𝐵))
22	20, 21, 2	elab2 3635	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐺 ↔ 𝑛:{𝑌}⟶𝐵)
23	19, 22	anbi12i 640	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺) ↔ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵))
24		simp3 1156	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → (𝑚‘𝑌) = (𝑛‘𝑌))
25		simp1r 1217	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → 𝑌 ∈ 𝐴)
26		fveq2 6886	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑚‘𝑦) = (𝑚‘𝑌))
27		fveq2 6886	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑛‘𝑦) = (𝑛‘𝑌))
28	26, 27	eqeq12d 2776	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → ((𝑚‘𝑦) = (𝑛‘𝑦) ↔ (𝑚‘𝑌) = (𝑛‘𝑌)))
29	28	ralsng 4632	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ 𝐴 → (∀𝑦 ∈ {𝑌} (𝑚‘𝑦) = (𝑛‘𝑦) ↔ (𝑚‘𝑌) = (𝑛‘𝑌)))
30	25, 29	syl 18	. . . . . . . . . . 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 260	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → ∀𝑦 ∈ {𝑌} (𝑚‘𝑦) = (𝑛‘𝑦))
32		ffn 6710	. . . . . . . . . . . . 13 ⊢ (𝑚:{𝑌}⟶𝐵 → 𝑚 Fn {𝑌})
33		ffn 6710	. . . . . . . . . . . . 13 ⊢ (𝑛:{𝑌}⟶𝐵 → 𝑛 Fn {𝑌})
34	32, 33	anim12i 625	. . . . . . . . . . . 12 ⊢ ((𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) → (𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}))
35	34	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → (𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}))
36		eqfnfv 7030	. . . . . . . . . . 11 ⊢ ((𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}) → (𝑚 = 𝑛 ↔ ∀𝑦 ∈ {𝑌} (𝑚‘𝑦) = (𝑛‘𝑦)))
37	35, 36	syl 18	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → 𝑚 = 𝑛)
39	38	3exp 1137	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) → ((𝑚‘𝑌) = (𝑛‘𝑌) → 𝑚 = 𝑛)))
40	23, 39	biimtrid 245	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺) → ((𝑚‘𝑌) = (𝑛‘𝑌) → 𝑚 = 𝑛)))
41	40	imp 412	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝑚‘𝑌) = (𝑛‘𝑌) → 𝑚 = 𝑛))
42	16, 41	syld 48	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → (∀𝑎 ∈ 𝐴 (𝑚‘𝑌) = (𝑛‘𝑌) → 𝑚 = 𝑛))
43	13, 42	sylbid 243	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) → 𝑚 = 𝑛))
44	9, 43	sylbid 243	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝐻‘𝑚) = (𝐻‘𝑛) → 𝑚 = 𝑛))
45	44	ralrimivva 3205	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ∀𝑚 ∈ 𝐺 ∀𝑛 ∈ 𝐺 ((𝐻‘𝑚) = (𝐻‘𝑛) → 𝑚 = 𝑛))
46		dff13 7259	. 2 ⊢ (𝐻:𝐺–1-1→𝐹 ↔ (𝐻:𝐺⟶𝐹 ∧ ∀𝑚 ∈ 𝐺 ∀𝑛 ∈ 𝐺 ((𝐻‘𝑚) = (𝐻‘𝑛) → 𝑚 = 𝑛)))
47	4, 45, 46	sylanbrc 595	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1→𝐹)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450  {csn 4584   ↦ cmpt 5185   Fn wfn 6534  ⟶wf 6535  –1-1→wf1 6536  ‘cfv 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547

This theorem is used by:  cfsetsnfsetf1o  48225