Theorem cfsetsnfsetf1 47009

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 47008	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)
5	1, 2, 3	cfsetsnfsetfv 47007	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑚 ∈ 𝐺) → (𝐻‘𝑚) = (𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)))
6	5	ad2ant2r 747	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → (𝐻‘𝑚) = (𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)))
7	1, 2, 3	cfsetsnfsetfv 47007	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ 𝐺) → (𝐻‘𝑛) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
8	7	ad2ant2rl 749	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → (𝐻‘𝑛) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
9	6, 8	eqeq12d 2751	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝐻‘𝑚) = (𝐻‘𝑛) ↔ (𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌))))
10		fvexd 6922	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) ∧ 𝑎 ∈ 𝐴) → (𝑚‘𝑌) ∈ V)
11	10	ralrimiva 3144	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ∀𝑎 ∈ 𝐴 (𝑚‘𝑌) ∈ V)
12		mpteqb 7035	. . . . . 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 769	. . . . . . 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 3612	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → (∀𝑎 ∈ 𝐴 (𝑚‘𝑌) = (𝑛‘𝑌) → (𝑚‘𝑌) = (𝑛‘𝑌)))
17		vex 3482	. . . . . . . . . 10 ⊢ 𝑚 ∈ V
18		feq1 6717	. . . . . . . . . 10 ⊢ (𝑥 = 𝑚 → (𝑥:{𝑌}⟶𝐵 ↔ 𝑚:{𝑌}⟶𝐵))
19	17, 18, 2	elab2 3685	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝐺 ↔ 𝑚:{𝑌}⟶𝐵)
20		vex 3482	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
21		feq1 6717	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝑥:{𝑌}⟶𝐵 ↔ 𝑛:{𝑌}⟶𝐵))
22	20, 21, 2	elab2 3685	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐺 ↔ 𝑛:{𝑌}⟶𝐵)
23	19, 22	anbi12i 628	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺) ↔ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵))
24		simp3 1137	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → (𝑚‘𝑌) = (𝑛‘𝑌))
25		simp1r 1197	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → 𝑌 ∈ 𝐴)
26		fveq2 6907	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑚‘𝑦) = (𝑚‘𝑌))
27		fveq2 6907	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑌 → (𝑛‘𝑦) = (𝑛‘𝑌))
28	26, 27	eqeq12d 2751	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → ((𝑚‘𝑦) = (𝑛‘𝑦) ↔ (𝑚‘𝑌) = (𝑛‘𝑌)))
29	28	ralsng 4680	. . . . . . . . . . . 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 257	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → ∀𝑦 ∈ {𝑌} (𝑚‘𝑦) = (𝑛‘𝑦))
32		ffn 6737	. . . . . . . . . . . . 13 ⊢ (𝑚:{𝑌}⟶𝐵 → 𝑚 Fn {𝑌})
33		ffn 6737	. . . . . . . . . . . . 13 ⊢ (𝑛:{𝑌}⟶𝐵 → 𝑛 Fn {𝑌})
34	32, 33	anim12i 613	. . . . . . . . . . . 12 ⊢ ((𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) → (𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}))
35	34	3ad2ant2 1133	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → (𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}))
36		eqfnfv 7051	. . . . . . . . . . 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 257	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) ∧ (𝑚‘𝑌) = (𝑛‘𝑌)) → 𝑚 = 𝑛)
39	38	3exp 1118	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((𝑚:{𝑌}⟶𝐵 ∧ 𝑛:{𝑌}⟶𝐵) → ((𝑚‘𝑌) = (𝑛‘𝑌) → 𝑚 = 𝑛)))
40	23, 39	biimtrid 242	. . . . . . 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 240	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝑎 ∈ 𝐴 ↦ (𝑚‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) → 𝑚 = 𝑛))
44	9, 43	sylbid 240	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ (𝑚 ∈ 𝐺 ∧ 𝑛 ∈ 𝐺)) → ((𝐻‘𝑚) = (𝐻‘𝑛) → 𝑚 = 𝑛))
45	44	ralrimivva 3200	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ∀𝑚 ∈ 𝐺 ∀𝑛 ∈ 𝐺 ((𝐻‘𝑚) = (𝐻‘𝑛) → 𝑚 = 𝑛))
46		dff13 7275	. 2 ⊢ (𝐻:𝐺–1-1→𝐹 ↔ (𝐻:𝐺⟶𝐹 ∧ ∀𝑚 ∈ 𝐺 ∀𝑛 ∈ 𝐺 ((𝐻‘𝑚) = (𝐻‘𝑛) → 𝑚 = 𝑛)))
47	4, 45, 46	sylanbrc 583	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1→𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  ∃wrex 3068  Vcvv 3478  {csn 4631   ↦ cmpt 5231   Fn wfn 6558  ⟶wf 6559  –1-1→wf1 6560  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571

This theorem is referenced by:  cfsetsnfsetf1o  47011