Theorem cfsetsnfsetfo 44554

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 44552	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)
5		vex 3436	. . . . 5 ⊢ 𝑚 ∈ V
6		feq1 6581	. . . . . 6 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵))
7		fveq1 6773	. . . . . . . . . 10 ⊢ (𝑓 = 𝑚 → (𝑓‘𝑧) = (𝑚‘𝑧))
8	7	adantr 481	. . . . . . . . 9 ⊢ ((𝑓 = 𝑚 ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) = (𝑚‘𝑧))
9	8	eqeq1d 2740	. . . . . . . 8 ⊢ ((𝑓 = 𝑚 ∧ 𝑧 ∈ 𝐴) → ((𝑓‘𝑧) = 𝑏 ↔ (𝑚‘𝑧) = 𝑏))
10	9	ralbidva 3111	. . . . . . 7 ⊢ (𝑓 = 𝑚 → (∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏))
11	10	rexbidv 3226	. . . . . 6 ⊢ (𝑓 = 𝑚 → (∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏))
12	6, 11	anbi12d 631	. . . . 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 3613	. . . 4 ⊢ (𝑚 ∈ 𝐹 ↔ (𝑚:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏))
14		simpllr 773	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) ∧ 𝑦 ∈ {𝑌}) → 𝑏 ∈ 𝐵)
15		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑌} ↦ 𝑏) = (𝑦 ∈ {𝑌} ↦ 𝑏)
16	14, 15	fmptd 6988	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵)
17		snex 5354	. . . . . . . . . . . 12 ⊢ {𝑌} ∈ V
18	17	mptex 7099	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑌} ↦ 𝑏) ∈ V
19		feq1 6581	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑥:{𝑌}⟶𝐵 ↔ (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵))
20	18, 19, 2	elab2 3613	. . . . . . . . . 10 ⊢ ((𝑦 ∈ {𝑌} ↦ 𝑏) ∈ 𝐺 ↔ (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵)
21	16, 20	sylibr 233	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → (𝑦 ∈ {𝑌} ↦ 𝑏) ∈ 𝐺)
22		fveq1 6773	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑛‘𝑌) = ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))
23	22	mpteq2dv 5176	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)))
24	23	eqeq2d 2749	. . . . . . . . . 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 2739	. . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) ∧ 𝑦 = 𝑌) → 𝑏 = 𝑏)
30		snidg 4595	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ 𝐴 → 𝑌 ∈ {𝑌})
31	30	ad6antlr 734	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → 𝑌 ∈ {𝑌})
32		simpllr 773	. . . . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . 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 3108	. . . . . . . . . . 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 6600	. . . . . . . . . . . . . . 15 ⊢ (𝑚:𝐴⟶𝐵 → 𝑚 Fn 𝐴)
43	42	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → 𝑚 Fn 𝐴)
44		nfv 1917	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵)
45		fvexd 6789	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑎 ∈ 𝐴) → ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌) ∈ V)
46		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))
47	44, 45, 46	fnmptd 6574	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴)
48	43, 47	jca 512	. . . . . . . . . . . . 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 6909	. . . . . . . . . . 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 256	. . . . . . . . 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 3563	. . . . . . . 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 780	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → 𝐴 ∈ 𝑉)
56	1, 2, 3	cfsetsnfsetfv 44551	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ 𝐺) → (𝐻‘𝑛) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
57	55, 56	sylan 580	. . . . . . . . . 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 2749	. . . . . . . . 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 3225	. . . . . . . 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 256	. . . . . . 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 3213	. . . . 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	syl5bi 241	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑚 ∈ 𝐹 → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
65	64	ralrimiv 3102	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ∀𝑚 ∈ 𝐹 ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛))
66		dffo3 6978	. 2 ⊢ (𝐻:𝐺–onto→𝐹 ↔ (𝐻:𝐺⟶𝐹 ∧ ∀𝑚 ∈ 𝐹 ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
67	4, 65, 66	sylanbrc 583	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–onto→𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065  Vcvv 3432  {csn 4561   ↦ cmpt 5157   Fn wfn 6428  ⟶wf 6429  –onto→wfo 6431  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441

This theorem is referenced by:  cfsetsnfsetf1o  44555