Theorem cfsetsnfsetfo 46975

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 46973	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)
5		vex 3492	. . . . 5 ⊢ 𝑚 ∈ V
6		feq1 6728	. . . . . 6 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴⟶𝐵 ↔ 𝑚:𝐴⟶𝐵))
7		fveq1 6919	. . . . . . . . . 10 ⊢ (𝑓 = 𝑚 → (𝑓‘𝑧) = (𝑚‘𝑧))
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝑓 = 𝑚 ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) = (𝑚‘𝑧))
9	8	eqeq1d 2742	. . . . . . . 8 ⊢ ((𝑓 = 𝑚 ∧ 𝑧 ∈ 𝐴) → ((𝑓‘𝑧) = 𝑏 ↔ (𝑚‘𝑧) = 𝑏))
10	9	ralbidva 3182	. . . . . . 7 ⊢ (𝑓 = 𝑚 → (∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏 ↔ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏))
11	10	rexbidv 3185	. . . . . 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 3698	. . . 4 ⊢ (𝑚 ∈ 𝐹 ↔ (𝑚:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏))
14		simpllr 775	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) ∧ 𝑦 ∈ {𝑌}) → 𝑏 ∈ 𝐵)
15		eqid 2740	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑌} ↦ 𝑏) = (𝑦 ∈ {𝑌} ↦ 𝑏)
16	14, 15	fmptd 7148	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵)
17		snex 5451	. . . . . . . . . . . 12 ⊢ {𝑌} ∈ V
18	17	mptex 7260	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑌} ↦ 𝑏) ∈ V
19		feq1 6728	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑥:{𝑌}⟶𝐵 ↔ (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵))
20	18, 19, 2	elab2 3698	. . . . . . . . . 10 ⊢ ((𝑦 ∈ {𝑌} ↦ 𝑏) ∈ 𝐺 ↔ (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵)
21	16, 20	sylibr 234	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → (𝑦 ∈ {𝑌} ↦ 𝑏) ∈ 𝐺)
22		fveq1 6919	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑛‘𝑌) = ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))
23	22	mpteq2dv 5268	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)))
24	23	eqeq2d 2751	. . . . . . . . . 10 ⊢ (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑚 = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)) ↔ 𝑚 = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))))
25	24	adantl 481	. . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → (𝑚‘𝑧) = 𝑏)
27		eqidd 2741	. . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) ∧ 𝑦 = 𝑌) → 𝑏 = 𝑏)
30		snidg 4682	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ 𝐴 → 𝑌 ∈ {𝑌})
31	30	ad6antlr 736	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → 𝑌 ∈ {𝑌})
32		simpllr 775	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → 𝑏 ∈ 𝐵)
33	32	adantr 480	. . . . . . . . . . . . . . . 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 7036	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌) = 𝑏)
35		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
36	35	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ (𝑚‘𝑧) = 𝑏) → 𝑧 ∈ 𝐴)
37	27, 34, 36, 32	fvmptd 7036	. . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 3173	. . . . . . . . . . 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 406	. . . . . . . . . 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 6747	. . . . . . . . . . . . . . 15 ⊢ (𝑚:𝐴⟶𝐵 → 𝑚 Fn 𝐴)
43	42	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → 𝑚 Fn 𝐴)
44		nfv 1913	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵)
45		fvexd 6935	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑎 ∈ 𝐴) → ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌) ∈ V)
46		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) = (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))
47	44, 45, 46	fnmptd 6721	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴)
48	43, 47	jca 511	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → (𝑚 Fn 𝐴 ∧ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴))
49	48	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑚 Fn 𝐴 ∧ (𝑎 ∈ 𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴))
50	49	adantr 480	. . . . . . . . . . 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 7064	. . . . . . . . . . 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 257	. . . . . . . . 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 3637	. . . . . . . 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 782	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → 𝐴 ∈ 𝑉)
56	1, 2, 3	cfsetsnfsetfv 46972	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ 𝐺) → (𝐻‘𝑛) = (𝑎 ∈ 𝐴 ↦ (𝑛‘𝑌)))
57	55, 56	sylan 579	. . . . . . . . . 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 2751	. . . . . . . . 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 3183	. . . . . . . 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 257	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) ∧ ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛))
61	60	ex 412	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) ∧ 𝑏 ∈ 𝐵) → (∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏 → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
62	61	rexlimdva 3161	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) ∧ 𝑚:𝐴⟶𝐵) → (∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏 → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
63	62	expimpd 453	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ((𝑚:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑚‘𝑧) = 𝑏) → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
64	13, 63	biimtrid 242	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (𝑚 ∈ 𝐹 → ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
65	64	ralrimiv 3151	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → ∀𝑚 ∈ 𝐹 ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛))
66		dffo3 7136	. 2 ⊢ (𝐻:𝐺–onto→𝐹 ↔ (𝐻:𝐺⟶𝐹 ∧ ∀𝑚 ∈ 𝐹 ∃𝑛 ∈ 𝐺 𝑚 = (𝐻‘𝑛)))
67	4, 65, 66	sylanbrc 582	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–onto→𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488  {csn 4648   ↦ cmpt 5249   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  cfsetsnfsetf1o  46976