Theorem cfsetsnfsetfo 47531

Description: The mapping of the class of singleton functions into the class of constant functions is a surjection. (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
cfsetsnfsetfo	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–onto→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,𝑏   𝑓,𝑔,𝑧   𝑧,𝐵   𝑥,𝑏,𝑧   𝐻,𝑏

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

Proof of Theorem cfsetsnfsetfo

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 47529	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)
5		vex 3435	. . . . 5 ⊢ 𝑚 ∈ V
6		feq1 6634	. . . . . 6 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵))
7		fveq1 6827	. . . . . . . . . 10 ⊢ (𝑓 = 𝑚 → (𝑓‘𝑧) = (𝑚‘𝑧))
8	7	adantr 481	. . . . . . . . 9 ⊢ ((𝑓 = 𝑚 ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) = (𝑚‘𝑧))
9	8	eqeq1d 2741	. . . . . . . 8 ⊢ ((𝑓 = 𝑚 ∧ 𝑧 ∈ 𝐴) → ((𝑓‘𝑧) = 𝑏 ↔ (𝑚‘𝑧) = 𝑏))
10	9	ralbidva 3160	. . . . . . 7 ⊢ (𝑓 = 𝑚 → (∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏))
11	10	rexbidv 3163	. . . . . 6 ⊢ (𝑓 = 𝑚 → (∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏))
12	6, 11	anbi12d 638	. . . . 5 ⊢ (𝑓 = 𝑚 → ((𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏) ↔ (𝑚:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏)))
13	5, 12, 1	elab2 3620	. . . 4 ⊢ (𝑚 ∈ 𝐹 ↔ (𝑚:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏))
14		simpllr 781	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) ∧ 𝑦 ∈ {𝑌}) → 𝑏 ∈ 𝐵)
15		eqid 2739	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑌} ↦ 𝑏) = (𝑦 ∈ {𝑌} ↦ 𝑏)
16	14, 15	fmptd 7056	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵)
17		snex 5369	. . . . . . . . . . . 12 ⊢ {𝑌} ∈ V
18	17	mptex 7168	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑌} ↦ 𝑏) ∈ V
19		feq1 6634	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑥:{𝑌}⟶𝐵 ↔ (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵))
20	18, 19, 2	elab2 3620	. . . . . . . . . 10 ⊢ ((𝑦 ∈ {𝑌} ↦ 𝑏) ∈ 𝐺 ↔ (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵)
21	16, 20	sylibr 235	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → (𝑦 ∈ {𝑌} ↦ 𝑏) ∈ 𝐺)
22		fveq1 6827	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑛‘𝑌) = ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))
23	22	mpteq2dv 5167	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)))
24	23	eqeq2d 2750	. . . . . . . . . 10 ⊢ (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑚 = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) ↔ 𝑚 = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))))
25	24	adantl 482	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) ∧ 𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏)) → (𝑚 = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) ↔ 𝑚 = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))))
26		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → (𝑚‘𝑧) = 𝑏)
27		eqidd 2740	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)))
28		eqidd 2740	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → (𝑦 ∈ {𝑌} ↦ 𝑏) = (𝑦 ∈ {𝑌} ↦ 𝑏))
29		eqidd 2740	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) ∧ 𝑦 = 𝑌) → 𝑏 = 𝑏)
30		snidg 4593	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ 𝐴 → 𝑌 ∈ {𝑌})
31	30	ad6antlr 743	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → 𝑌 ∈ {𝑌})
32		simpllr 781	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → 𝑏 ∈ 𝐵)
33	32	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → 𝑏 ∈ 𝐵)
34	28, 29, 31, 33	fvmptd 6944	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌) = 𝑏)
35		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
36	35	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → 𝑧 ∈ 𝐴)
37	27, 34, 36, 32	fvmptd 6944	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → ((𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧) = 𝑏)
38	26, 37	eqtr4d 2777	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → (𝑚‘𝑧) = ((𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧))
39	38	ex 413	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) → ((𝑚‘𝑧) = 𝑏 → (𝑚‘𝑧) = ((𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧)))
40	39	ralimdva 3151	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) → (∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏 → ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = ((𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧)))
41	40	imp 407	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = ((𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧))
42		ffn 6656	. . . . . . . . . . . . . . 15 ⊢ (𝑚:𝐴⟶𝐵 → 𝑚 Fn 𝐴)
43	42	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → 𝑚 Fn 𝐴)
44		nfv 1921	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵)
45		fvexd 6843	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑎 ∈ 𝐴) → ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌) ∈ V)
46		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))
47	44, 45, 46	fnmptd 6627	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴)
48	43, 47	jca 516	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → (𝑚 Fn 𝐴 ∧ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴))
49	48	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑚 Fn 𝐴 ∧ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴))
50	49	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → (𝑚 Fn 𝐴 ∧ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴))
51		eqfnfv 6972	. . . . . . . . . . 11 ⊢ ((𝑚 Fn 𝐴 ∧ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴) → (𝑚 = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) ↔ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = ((𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧)))
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → (𝑚 = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) ↔ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = ((𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧)))
53	41, 52	mpbird 258	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → 𝑚 = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)))
54	21, 25, 53	rspcedvd 3562	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → ∃𝑛 ∈ 𝐺 𝑚 = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
55		simp-4l 788	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → 𝐴 ∈ 𝑉)
56	1, 2, 3	cfsetsnfsetfv 47528	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ 𝐺) → (𝐻‘𝑛) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
57	55, 56	sylan 586	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) ∧ 𝑛 ∈ 𝐺) → (𝐻‘𝑛) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
58	57	eqeq2d 2750	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) ∧ 𝑛 ∈ 𝐺) → (𝑚 = (𝐻‘𝑛) ↔ 𝑚 = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌))))
59	58	rexbidva 3161	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → (∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛) ↔ ∃𝑛 ∈ 𝐺 𝑚 = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌))))
60	54, 59	mpbird 258	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛))
61	60	ex 413	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) → (∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏 → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
62	61	rexlimdva 3140	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → (∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏 → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
63	62	expimpd 454	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((𝑚:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
64	13, 63	biimtrid 243	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑚 ∈ 𝐹 → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
65	64	ralrimiv 3130	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ∀𝑚 ∈ 𝐹 ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛))
66		dffo3 7044	. 2 ⊢ (𝐻:𝐺–onto→𝐹 ↔ (𝐻:𝐺⟶𝐹 ∧ ∀𝑚 ∈ 𝐹 ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
67	4, 65, 66	sylanbrc 589	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–onto→𝐹)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  Vcvv 3431  {csn 4556   ↦ cmpt 5154   Fn wfn 6481  ⟶wf 6482  –onto→wfo 6484  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494

This theorem is referenced by:  cfsetsnfsetf1o  47532