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Theorem cfsetsnfsetfo 44241
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:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)}
cfsetsnfsetfv.g 𝐺 = {𝑥𝑥:{𝑌}⟶𝐵}
cfsetsnfsetfv.h 𝐻 = (𝑔𝐺 ↦ (𝑎𝐴 ↦ (𝑔𝑌)))
Assertion
Ref Expression
cfsetsnfsetfo ((𝐴𝑉𝑌𝐴) → 𝐻:𝐺onto𝐹)
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.
StepHypRef Expression
1 cfsetsnfsetfv.f . . 3 𝐹 = {𝑓 ∣ (𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏)}
2 cfsetsnfsetfv.g . . 3 𝐺 = {𝑥𝑥:{𝑌}⟶𝐵}
3 cfsetsnfsetfv.h . . 3 𝐻 = (𝑔𝐺 ↦ (𝑎𝐴 ↦ (𝑔𝑌)))
41, 2, 3cfsetsnfsetf 44239 . 2 ((𝐴𝑉𝑌𝐴) → 𝐻:𝐺𝐹)
5 vex 3419 . . . . 5 𝑚 ∈ V
6 feq1 6535 . . . . . 6 (𝑓 = 𝑚 → (𝑓:𝐴𝐵𝑚:𝐴𝐵))
7 fveq1 6725 . . . . . . . . . 10 (𝑓 = 𝑚 → (𝑓𝑧) = (𝑚𝑧))
87adantr 484 . . . . . . . . 9 ((𝑓 = 𝑚𝑧𝐴) → (𝑓𝑧) = (𝑚𝑧))
98eqeq1d 2740 . . . . . . . 8 ((𝑓 = 𝑚𝑧𝐴) → ((𝑓𝑧) = 𝑏 ↔ (𝑚𝑧) = 𝑏))
109ralbidva 3118 . . . . . . 7 (𝑓 = 𝑚 → (∀𝑧𝐴 (𝑓𝑧) = 𝑏 ↔ ∀𝑧𝐴 (𝑚𝑧) = 𝑏))
1110rexbidv 3223 . . . . . 6 (𝑓 = 𝑚 → (∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏 ↔ ∃𝑏𝐵𝑧𝐴 (𝑚𝑧) = 𝑏))
126, 11anbi12d 634 . . . . 5 (𝑓 = 𝑚 → ((𝑓:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑓𝑧) = 𝑏) ↔ (𝑚:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑚𝑧) = 𝑏)))
135, 12, 1elab2 3598 . . . 4 (𝑚𝐹 ↔ (𝑚:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑚𝑧) = 𝑏))
14 simpllr 776 . . . . . . . . . . 11 ((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) ∧ 𝑦 ∈ {𝑌}) → 𝑏𝐵)
15 eqid 2738 . . . . . . . . . . 11 (𝑦 ∈ {𝑌} ↦ 𝑏) = (𝑦 ∈ {𝑌} ↦ 𝑏)
1614, 15fmptd 6940 . . . . . . . . . 10 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) → (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵)
17 snex 5333 . . . . . . . . . . . 12 {𝑌} ∈ V
1817mptex 7048 . . . . . . . . . . 11 (𝑦 ∈ {𝑌} ↦ 𝑏) ∈ V
19 feq1 6535 . . . . . . . . . . 11 (𝑥 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑥:{𝑌}⟶𝐵 ↔ (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵))
2018, 19, 2elab2 3598 . . . . . . . . . 10 ((𝑦 ∈ {𝑌} ↦ 𝑏) ∈ 𝐺 ↔ (𝑦 ∈ {𝑌} ↦ 𝑏):{𝑌}⟶𝐵)
2116, 20sylibr 237 . . . . . . . . 9 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) → (𝑦 ∈ {𝑌} ↦ 𝑏) ∈ 𝐺)
22 fveq1 6725 . . . . . . . . . . . 12 (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑛𝑌) = ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))
2322mpteq2dv 5160 . . . . . . . . . . 11 (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑎𝐴 ↦ (𝑛𝑌)) = (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)))
2423eqeq2d 2749 . . . . . . . . . 10 (𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏) → (𝑚 = (𝑎𝐴 ↦ (𝑛𝑌)) ↔ 𝑚 = (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))))
2524adantl 485 . . . . . . . . 9 ((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) ∧ 𝑛 = (𝑦 ∈ {𝑌} ↦ 𝑏)) → (𝑚 = (𝑎𝐴 ↦ (𝑛𝑌)) ↔ 𝑚 = (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))))
26 simpr 488 . . . . . . . . . . . . . 14 ((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) ∧ (𝑚𝑧) = 𝑏) → (𝑚𝑧) = 𝑏)
27 eqidd 2739 . . . . . . . . . . . . . . 15 ((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) ∧ (𝑚𝑧) = 𝑏) → (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) = (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)))
28 eqidd 2739 . . . . . . . . . . . . . . . 16 (((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) ∧ (𝑚𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → (𝑦 ∈ {𝑌} ↦ 𝑏) = (𝑦 ∈ {𝑌} ↦ 𝑏))
29 eqidd 2739 . . . . . . . . . . . . . . . 16 ((((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) ∧ (𝑚𝑧) = 𝑏) ∧ 𝑎 = 𝑧) ∧ 𝑦 = 𝑌) → 𝑏 = 𝑏)
30 snidg 4584 . . . . . . . . . . . . . . . . 17 (𝑌𝐴𝑌 ∈ {𝑌})
3130ad6antlr 737 . . . . . . . . . . . . . . . 16 (((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) ∧ (𝑚𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → 𝑌 ∈ {𝑌})
32 simpllr 776 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) ∧ (𝑚𝑧) = 𝑏) → 𝑏𝐵)
3332adantr 484 . . . . . . . . . . . . . . . 16 (((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) ∧ (𝑚𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → 𝑏𝐵)
3428, 29, 31, 33fvmptd 6834 . . . . . . . . . . . . . . 15 (((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) ∧ (𝑚𝑧) = 𝑏) ∧ 𝑎 = 𝑧) → ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌) = 𝑏)
35 simpr 488 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) → 𝑧𝐴)
3635adantr 484 . . . . . . . . . . . . . . 15 ((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) ∧ (𝑚𝑧) = 𝑏) → 𝑧𝐴)
3727, 34, 36, 32fvmptd 6834 . . . . . . . . . . . . . 14 ((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) ∧ (𝑚𝑧) = 𝑏) → ((𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧) = 𝑏)
3826, 37eqtr4d 2781 . . . . . . . . . . . . 13 ((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) ∧ (𝑚𝑧) = 𝑏) → (𝑚𝑧) = ((𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧))
3938ex 416 . . . . . . . . . . . 12 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ 𝑧𝐴) → ((𝑚𝑧) = 𝑏 → (𝑚𝑧) = ((𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧)))
4039ralimdva 3101 . . . . . . . . . . 11 ((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) → (∀𝑧𝐴 (𝑚𝑧) = 𝑏 → ∀𝑧𝐴 (𝑚𝑧) = ((𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧)))
4140imp 410 . . . . . . . . . 10 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) → ∀𝑧𝐴 (𝑚𝑧) = ((𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧))
42 ffn 6554 . . . . . . . . . . . . . . 15 (𝑚:𝐴𝐵𝑚 Fn 𝐴)
4342adantl 485 . . . . . . . . . . . . . 14 (((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) → 𝑚 Fn 𝐴)
44 nfv 1922 . . . . . . . . . . . . . . 15 𝑎((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵)
45 fvexd 6741 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑎𝐴) → ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌) ∈ V)
46 eqid 2738 . . . . . . . . . . . . . . 15 (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) = (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))
4744, 45, 46fnmptd 6528 . . . . . . . . . . . . . 14 (((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) → (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴)
4843, 47jca 515 . . . . . . . . . . . . 13 (((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) → (𝑚 Fn 𝐴 ∧ (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴))
4948adantr 484 . . . . . . . . . . . 12 ((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) → (𝑚 Fn 𝐴 ∧ (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴))
5049adantr 484 . . . . . . . . . . 11 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) → (𝑚 Fn 𝐴 ∧ (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴))
51 eqfnfv 6861 . . . . . . . . . . 11 ((𝑚 Fn 𝐴 ∧ (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) Fn 𝐴) → (𝑚 = (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) ↔ ∀𝑧𝐴 (𝑚𝑧) = ((𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧)))
5250, 51syl 17 . . . . . . . . . 10 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) → (𝑚 = (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)) ↔ ∀𝑧𝐴 (𝑚𝑧) = ((𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌))‘𝑧)))
5341, 52mpbird 260 . . . . . . . . 9 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) → 𝑚 = (𝑎𝐴 ↦ ((𝑦 ∈ {𝑌} ↦ 𝑏)‘𝑌)))
5421, 25, 53rspcedvd 3547 . . . . . . . 8 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) → ∃𝑛𝐺 𝑚 = (𝑎𝐴 ↦ (𝑛𝑌)))
55 simp-4l 783 . . . . . . . . . . 11 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) → 𝐴𝑉)
561, 2, 3cfsetsnfsetfv 44238 . . . . . . . . . . 11 ((𝐴𝑉𝑛𝐺) → (𝐻𝑛) = (𝑎𝐴 ↦ (𝑛𝑌)))
5755, 56sylan 583 . . . . . . . . . 10 ((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) ∧ 𝑛𝐺) → (𝐻𝑛) = (𝑎𝐴 ↦ (𝑛𝑌)))
5857eqeq2d 2749 . . . . . . . . 9 ((((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) ∧ 𝑛𝐺) → (𝑚 = (𝐻𝑛) ↔ 𝑚 = (𝑎𝐴 ↦ (𝑛𝑌))))
5958rexbidva 3222 . . . . . . . 8 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) → (∃𝑛𝐺 𝑚 = (𝐻𝑛) ↔ ∃𝑛𝐺 𝑚 = (𝑎𝐴 ↦ (𝑛𝑌))))
6054, 59mpbird 260 . . . . . . 7 (((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) ∧ ∀𝑧𝐴 (𝑚𝑧) = 𝑏) → ∃𝑛𝐺 𝑚 = (𝐻𝑛))
6160ex 416 . . . . . 6 ((((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) ∧ 𝑏𝐵) → (∀𝑧𝐴 (𝑚𝑧) = 𝑏 → ∃𝑛𝐺 𝑚 = (𝐻𝑛)))
6261rexlimdva 3210 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ 𝑚:𝐴𝐵) → (∃𝑏𝐵𝑧𝐴 (𝑚𝑧) = 𝑏 → ∃𝑛𝐺 𝑚 = (𝐻𝑛)))
6362expimpd 457 . . . 4 ((𝐴𝑉𝑌𝐴) → ((𝑚:𝐴𝐵 ∧ ∃𝑏𝐵𝑧𝐴 (𝑚𝑧) = 𝑏) → ∃𝑛𝐺 𝑚 = (𝐻𝑛)))
6413, 63syl5bi 245 . . 3 ((𝐴𝑉𝑌𝐴) → (𝑚𝐹 → ∃𝑛𝐺 𝑚 = (𝐻𝑛)))
6564ralrimiv 3105 . 2 ((𝐴𝑉𝑌𝐴) → ∀𝑚𝐹𝑛𝐺 𝑚 = (𝐻𝑛))
66 dffo3 6930 . 2 (𝐻:𝐺onto𝐹 ↔ (𝐻:𝐺𝐹 ∧ ∀𝑚𝐹𝑛𝐺 𝑚 = (𝐻𝑛)))
674, 65, 66sylanbrc 586 1 ((𝐴𝑉𝑌𝐴) → 𝐻:𝐺onto𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  {cab 2715  wral 3062  wrex 3063  Vcvv 3415  {csn 4550  cmpt 5144   Fn wfn 6384  wf 6385  ontowfo 6387  cfv 6389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5188  ax-sep 5201  ax-nul 5208  ax-pr 5331
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3417  df-sbc 3704  df-csb 3821  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4829  df-iun 4915  df-br 5063  df-opab 5125  df-mpt 5145  df-id 5464  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-res 5572  df-ima 5573  df-iota 6347  df-fun 6391  df-fn 6392  df-f 6393  df-f1 6394  df-fo 6395  df-f1o 6396  df-fv 6397
This theorem is referenced by:  cfsetsnfsetf1o  44242
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