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Theorem cfsetsnfsetf1 45283
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 45282 . 2 ((𝐴𝑉𝑌𝐴) → 𝐻:𝐺𝐹)
51, 2, 3cfsetsnfsetfv 45281 . . . . . 6 ((𝐴𝑉𝑚𝐺) → (𝐻𝑚) = (𝑎𝐴 ↦ (𝑚𝑌)))
65ad2ant2r 745 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → (𝐻𝑚) = (𝑎𝐴 ↦ (𝑚𝑌)))
71, 2, 3cfsetsnfsetfv 45281 . . . . . 6 ((𝐴𝑉𝑛𝐺) → (𝐻𝑛) = (𝑎𝐴 ↦ (𝑛𝑌)))
87ad2ant2rl 747 . . . . 5 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → (𝐻𝑛) = (𝑎𝐴 ↦ (𝑛𝑌)))
96, 8eqeq12d 2752 . . . 4 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → ((𝐻𝑚) = (𝐻𝑛) ↔ (𝑎𝐴 ↦ (𝑚𝑌)) = (𝑎𝐴 ↦ (𝑛𝑌))))
10 fvexd 6857 . . . . . . 7 ((((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) ∧ 𝑎𝐴) → (𝑚𝑌) ∈ V)
1110ralrimiva 3143 . . . . . 6 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → ∀𝑎𝐴 (𝑚𝑌) ∈ V)
12 mpteqb 6967 . . . . . 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 767 . . . . . . 7 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → 𝑌𝐴)
15 idd 24 . . . . . . 7 ((((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) ∧ 𝑎 = 𝑌) → ((𝑚𝑌) = (𝑛𝑌) → (𝑚𝑌) = (𝑛𝑌)))
1614, 15rspcimdv 3571 . . . . . 6 (((𝐴𝑉𝑌𝐴) ∧ (𝑚𝐺𝑛𝐺)) → (∀𝑎𝐴 (𝑚𝑌) = (𝑛𝑌) → (𝑚𝑌) = (𝑛𝑌)))
17 vex 3449 . . . . . . . . . 10 𝑚 ∈ V
18 feq1 6649 . . . . . . . . . 10 (𝑥 = 𝑚 → (𝑥:{𝑌}⟶𝐵𝑚:{𝑌}⟶𝐵))
1917, 18, 2elab2 3634 . . . . . . . . 9 (𝑚𝐺𝑚:{𝑌}⟶𝐵)
20 vex 3449 . . . . . . . . . 10 𝑛 ∈ V
21 feq1 6649 . . . . . . . . . 10 (𝑥 = 𝑛 → (𝑥:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵))
2220, 21, 2elab2 3634 . . . . . . . . 9 (𝑛𝐺𝑛:{𝑌}⟶𝐵)
2319, 22anbi12i 627 . . . . . . . 8 ((𝑚𝐺𝑛𝐺) ↔ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵))
24 simp3 1138 . . . . . . . . . . 11 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → (𝑚𝑌) = (𝑛𝑌))
25 simp1r 1198 . . . . . . . . . . . 12 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → 𝑌𝐴)
26 fveq2 6842 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑚𝑦) = (𝑚𝑌))
27 fveq2 6842 . . . . . . . . . . . . . 14 (𝑦 = 𝑌 → (𝑛𝑦) = (𝑛𝑌))
2826, 27eqeq12d 2752 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → ((𝑚𝑦) = (𝑛𝑦) ↔ (𝑚𝑌) = (𝑛𝑌)))
2928ralsng 4634 . . . . . . . . . . . 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 256 . . . . . . . . . 10 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → ∀𝑦 ∈ {𝑌} (𝑚𝑦) = (𝑛𝑦))
32 ffn 6668 . . . . . . . . . . . . 13 (𝑚:{𝑌}⟶𝐵𝑚 Fn {𝑌})
33 ffn 6668 . . . . . . . . . . . . 13 (𝑛:{𝑌}⟶𝐵𝑛 Fn {𝑌})
3432, 33anim12i 613 . . . . . . . . . . . 12 ((𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) → (𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}))
35343ad2ant2 1134 . . . . . . . . . . 11 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → (𝑚 Fn {𝑌} ∧ 𝑛 Fn {𝑌}))
36 eqfnfv 6982 . . . . . . . . . . 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 256 . . . . . . . . 9 (((𝐴𝑉𝑌𝐴) ∧ (𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) ∧ (𝑚𝑌) = (𝑛𝑌)) → 𝑚 = 𝑛)
39383exp 1119 . . . . . . . 8 ((𝐴𝑉𝑌𝐴) → ((𝑚:{𝑌}⟶𝐵𝑛:{𝑌}⟶𝐵) → ((𝑚𝑌) = (𝑛𝑌) → 𝑚 = 𝑛)))
4023, 39biimtrid 241 . . . . . . 7 ((𝐴𝑉𝑌𝐴) → ((𝑚𝐺𝑛𝐺) → ((𝑚𝑌) = (𝑛𝑌) → 𝑚 = 𝑛)))
4140imp 407 . . . . . 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 3197 . 2 ((𝐴𝑉𝑌𝐴) → ∀𝑚𝐺𝑛𝐺 ((𝐻𝑚) = (𝐻𝑛) → 𝑚 = 𝑛))
46 dff13 7202 . 2 (𝐻:𝐺1-1𝐹 ↔ (𝐻:𝐺𝐹 ∧ ∀𝑚𝐺𝑛𝐺 ((𝐻𝑚) = (𝐻𝑛) → 𝑚 = 𝑛)))
474, 45, 46sylanbrc 583 1 ((𝐴𝑉𝑌𝐴) → 𝐻:𝐺1-1𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wral 3064  wrex 3073  Vcvv 3445  {csn 4586  cmpt 5188   Fn wfn 6491  wf 6492  1-1wf1 6493  cfv 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504
This theorem is referenced by:  cfsetsnfsetf1o  45285
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