Theorem cfsetsnfsetfo 47956

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 47954	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)
5		vex 3457	. . . . 5 ⊢ 𝑚 ∈ V
6		feq1 6684	. . . . . 6 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵))
7		fveq1 6881	. . . . . . . . . 10 ⊢ (𝑓 = 𝑚 → (𝑓‘𝑧) = (𝑚‘𝑧))
8	7	adantr 486	. . . . . . . . 9 ⊢ ((𝑓 = 𝑚 ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) = (𝑚‘𝑧))
9	8	eqeq1d 2764	. . . . . . . 8 ⊢ ((𝑓 = 𝑚 ∧ 𝑧 ∈ 𝐴) → ((𝑓‘𝑧) = 𝑏 ↔ (𝑚‘𝑧) = 𝑏))
10	9	ralbidva 3185	. . . . . . 7 ⊢ (𝑓 = 𝑚 → (∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏))
11	10	rexbidv 3188	. . . . . 6 ⊢ (𝑓 = 𝑚 → (∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏))
12	6, 11	anbi12d 644	. . . . 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 3639	. . . 4 ⊢ (𝑚 ∈ 𝐹 ↔ (𝑚:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏))
14		simpllr 788	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) ∧ 𝑦 ∈ {𝑌}) → 𝑏 ∈ 𝐵)
15		eqid 2762	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑌} ↦ 𝑏) = (𝑦 ∈ {𝑌} ↦ 𝑏)
16	14, 15	fmptd 7111	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵)
17		snex 5408	. . . . . . . . . . . 12 ⊢ {𝑌} ∈ V
18	17	mptex 7226	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑌} ↦ 𝑏) ∈ V
19		feq1 6684	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑥:{𝑌}⟶𝐵 ↔ (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵))
20	18, 19, 2	elab2 3639	. . . . . . . . . 10 ⊢ ((𝑦 ∈ {𝑌} ↦ 𝑏) ∈ 𝐺 ↔ (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵)
21	16, 20	sylibr 237	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → (𝑦 ∈ {𝑌} ↦ 𝑏) ∈ 𝐺)
22		fveq1 6881	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑛‘𝑌) = ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))
23	22	mpteq2dv 5203	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)))
24	23	eqeq2d 2773	. . . . . . . . . 10 ⊢ (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑚 = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) ↔ 𝑚 = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))))
25	24	adantl 487	. . . . . . . . 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 490	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → (𝑚‘𝑧) = 𝑏)
27		eqidd 2763	. . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) ∧ 𝑦 = 𝑌) → 𝑏 = 𝑏)
30		snidg 4624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ 𝐴 → 𝑌 ∈ {𝑌})
31	30	ad6antlr 750	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → 𝑌 ∈ {𝑌})
32		simpllr 788	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → 𝑏 ∈ 𝐵)
33	32	adantr 486	. . . . . . . . . . . . . . . 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 6998	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌) = 𝑏)
35		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
36	35	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → 𝑧 ∈ 𝐴)
37	27, 34, 36, 32	fvmptd 6998	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . 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 418	. . . . . . . . . . . 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 3176	. . . . . . . . . . 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 412	. . . . . . . . . 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 6706	. . . . . . . . . . . . . . 15 ⊢ (𝑚:𝐴⟶𝐵 → 𝑚 Fn 𝐴)
43	42	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → 𝑚 Fn 𝐴)
44		nfv 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵)
45		fvexd 6897	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑎 ∈ 𝐴) → ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌) ∈ V)
46		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))
47	44, 45, 46	fnmptd 6677	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴)
48	43, 47	jca 521	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → (𝑚 Fn 𝐴 ∧ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴))
49	48	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑚 Fn 𝐴 ∧ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴))
50	49	adantr 486	. . . . . . . . . . 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 7026	. . . . . . . . . . 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 18	. . . . . . . . . 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 260	. . . . . . . . 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 3581	. . . . . . . 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 795	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → 𝐴 ∈ 𝑉)
56	1, 2, 3	cfsetsnfsetfv 47953	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ 𝐺) → (𝐻‘𝑛) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
57	55, 56	sylan 592	. . . . . . . . . 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 2773	. . . . . . . . 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 3186	. . . . . . . 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 260	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛))
61	60	ex 418	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) → (∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏 → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
62	61	rexlimdva 3165	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → (∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏 → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
63	62	expimpd 459	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((𝑚:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
64	13, 63	biimtrid 245	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑚 ∈ 𝐹 → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
65	64	ralrimiv 3155	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ∀𝑚 ∈ 𝐹 ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛))
66		dffo3 7099	. 2 ⊢ (𝐻:𝐺–onto→𝐹 ↔ (𝐻:𝐺⟶𝐹 ∧ ∀𝑚 ∈ 𝐹 ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
67	4, 65, 66	sylanbrc 595	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–onto→𝐹)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∀wral 3078  ∃wrex 3088  Vcvv 3453  {csn 4587   ↦ cmpt 5190   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537

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-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is used by:  cfsetsnfsetf1o  47957