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Theorem cfsetsnfsetf1 45769
Description: The mapping of the class of singleton functions into the class of constant functions is an injection. (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
cfsetsnfsetf1 ((𝐴𝑉𝑌𝐴) → 𝐻:𝐺1-1𝐹)
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   𝑔,𝑏,𝑓,𝑧
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑧,𝑔)   𝐹(𝑥,𝑧,𝑓,𝑎,𝑏)   𝐺(𝑥,𝑓)   𝐻(𝑥,𝑧,𝑓,𝑔,𝑎,𝑏)   𝑉(𝑥,𝑓)

Proof of Theorem cfsetsnfsetf1
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 45768 . 2 ((𝐴𝑉𝑌𝐴) → 𝐻:𝐺𝐹)
51, 2, 3cfsetsnfsetfv 45767 . . . . . 6 ((𝐴𝑉𝑚𝐺) → (𝐻𝑚) = (𝑎𝐴 ↦ (𝑚𝑌)))
65ad2ant2r 746 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → (𝐻𝑚) = (𝑎𝐴 ↦ (𝑚𝑌)))
71, 2, 3cfsetsnfsetfv 45767 . . . . . 6 ((𝐴𝑉𝑛𝐺) → (𝐻𝑛) = (𝑎𝐴 ↦ (𝑛𝑌)))
87ad2ant2rl 748 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → (𝐻𝑛) = (𝑎𝐴 ↦ (𝑛𝑌)))
96, 8eqeq12d 2749 . . . 4 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → ((𝐻𝑚) = (𝐻𝑛) ↔ (𝑎𝐴 ↦ (𝑚𝑌)) = (𝑎𝐴 ↦ (𝑛𝑌))))
10 fvexd 6907 . . . . . . 7 ((((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) ∧ 𝑎𝐴) → (𝑚𝑌) ∈ V)
1110ralrimiva 3147 . . . . . 6 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → ∀𝑎𝐴 (𝑚𝑌) ∈ V)
12 mpteqb 7018 . . . . . 6 (∀𝑎𝐴 (𝑚𝑌) ∈ V → ((𝑎𝐴 ↦ (𝑚𝑌)) = (𝑎𝐴 ↦ (𝑛𝑌)) ↔ ∀𝑎𝐴 (𝑚𝑌) = (𝑛𝑌)))
1311, 12syl 17 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → ((𝑎𝐴 ↦ (𝑚𝑌)) = (𝑎𝐴 ↦ (𝑛𝑌)) ↔ ∀𝑎𝐴 (𝑚𝑌) = (𝑛𝑌)))
14 simplr 768 . . . . . . 7 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → 𝑌𝐴)
15 idd 24 . . . . . . 7 ((((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) ∧ 𝑎 = 𝑌) → ((𝑚𝑌) = (𝑛𝑌) → (𝑚𝑌) = (𝑛𝑌)))
1614, 15rspcimdv 3603 . . . . . 6 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → (∀𝑎𝐴 (𝑚𝑌) = (𝑛𝑌) → (𝑚𝑌) = (𝑛𝑌)))
17 vex 3479 . . . . . . . . . 10 𝑚 ∈ V
18 feq1 6699 . . . . . . . . . 10 (𝑥 = 𝑚 → (𝑥:{𝑌}⟶𝐵𝑚:{𝑌}⟶𝐵))
1917, 18, 2elab2 3673 . . . . . . . . 9 (𝑚𝐺𝑚:{𝑌}⟶𝐵)
20 vex 3479 . . . . . . . . . 10 𝑛 ∈ V
21 feq1 6699 . . . . . . . . . 10 (𝑥 = 𝑛 → (𝑥:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵))
2220, 21, 2elab2 3673 . . . . . . . . 9 (𝑛𝐺𝑛:{𝑌}⟶𝐵)
2319, 22anbi12i 628 . . . . . . . 8 ((𝑚𝐺𝑛𝐺) ↔ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵))
24 simp3 1139 . . . . . . . . . . 11 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → (𝑚𝑌) = (𝑛𝑌))
25 simp1r 1199 . . . . . . . . . . . 12 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → 𝑌𝐴)
26 fveq2 6892 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑚𝑦) = (𝑚𝑌))
27 fveq2 6892 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑛𝑦) = (𝑛𝑌))
2826, 27eqeq12d 2749 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑚𝑦) = (𝑛𝑦) ↔ (𝑚𝑌) = (𝑛𝑌)))
2928ralsng 4678 . . . . . . . . . . . 12 (𝑌𝐴 → (∀𝑦 ∈ {𝑌} (𝑚𝑦) = (𝑛𝑦) ↔ (𝑚𝑌) = (𝑛𝑌)))
3025, 29syl 17 . . . . . . . . . . 11 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → (∀𝑦 ∈ {𝑌} (𝑚𝑦) = (𝑛𝑦) ↔ (𝑚𝑌) = (𝑛𝑌)))
3124, 30mpbird 257 . . . . . . . . . 10 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → ∀𝑦 ∈ {𝑌} (𝑚𝑦) = (𝑛𝑦))
32 ffn 6718 . . . . . . . . . . . . 13 (𝑚:{𝑌}⟶𝐵𝑚 Fn {𝑌})
33 ffn 6718 . . . . . . . . . . . . 13 (𝑛:{𝑌}⟶𝐵𝑛 Fn {𝑌})
3432, 33anim12i 614 . . . . . . . . . . . 12 ((𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) → (𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}))
35343ad2ant2 1135 . . . . . . . . . . 11 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → (𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}))
36 eqfnfv 7033 . . . . . . . . . . 11 ((𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}) → (𝑚 = 𝑛 ↔ ∀𝑦 ∈ {𝑌} (𝑚𝑦) = (𝑛𝑦)))
3735, 36syl 17 . . . . . . . . . 10 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → (𝑚 = 𝑛 ↔ ∀𝑦 ∈ {𝑌} (𝑚𝑦) = (𝑛𝑦)))
3831, 37mpbird 257 . . . . . . . . 9 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → 𝑚 = 𝑛)
39383exp 1120 . . . . . . . 8 ((𝐴𝑉𝑌𝐴) → ((𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) → ((𝑚𝑌) = (𝑛𝑌) → 𝑚 = 𝑛)))
4023, 39biimtrid 241 . . . . . . 7 ((𝐴𝑉𝑌𝐴) → ((𝑚𝐺𝑛𝐺) → ((𝑚𝑌) = (𝑛𝑌) → 𝑚 = 𝑛)))
4140imp 408 . . . . . 6 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → ((𝑚𝑌) = (𝑛𝑌) → 𝑚 = 𝑛))
4216, 41syld 47 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → (∀𝑎𝐴 (𝑚𝑌) = (𝑛𝑌) → 𝑚 = 𝑛))
4313, 42sylbid 239 . . . 4 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → ((𝑎𝐴 ↦ (𝑚𝑌)) = (𝑎𝐴 ↦ (𝑛𝑌)) → 𝑚 = 𝑛))
449, 43sylbid 239 . . 3 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → ((𝐻𝑚) = (𝐻𝑛) → 𝑚 = 𝑛))
4544ralrimivva 3201 . 2 ((𝐴𝑉𝑌𝐴) → ∀𝑚𝐺𝑛𝐺 ((𝐻𝑚) = (𝐻𝑛) → 𝑚 = 𝑛))
46 dff13 7254 . 2 (𝐻:𝐺1-1𝐹 ↔ (𝐻:𝐺𝐹 ∧ ∀𝑚𝐺𝑛𝐺 ((𝐻𝑚) = (𝐻𝑛) → 𝑚 = 𝑛)))
474, 45, 46sylanbrc 584 1 ((𝐴𝑉𝑌𝐴) → 𝐻:𝐺1-1𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3062  wrex 3071  Vcvv 3475  {csn 4629  cmpt 5232   Fn wfn 6539  wf 6540  1-1wf1 6541  cfv 6544
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552
This theorem is referenced by:  cfsetsnfsetf1o  45771
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