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Theorem fsetfocdm 8757
Description: The class of functions with a given domain that is a set and a given codomain is mapped, through evaluation at a point of the domain, onto the codomain. (Contributed by AV, 15-Sep-2024.)
Hypotheses
Ref Expression
fsetfocdm.f 𝐹 = {𝑓𝑓:𝐴𝐵}
fsetfocdm.s 𝑆 = (𝑔𝐹 ↦ (𝑔𝑋))
Assertion
Ref Expression
fsetfocdm ((𝐴𝑉𝑋𝐴) → 𝑆:𝐹onto𝐵)
Distinct variable groups:   𝐴,𝑓,𝑔   𝐵,𝑓,𝑔   𝑔,𝐹   𝑔,𝑋   𝑓,𝐹   𝑆,𝑔   𝑓,𝑉,𝑔   𝑓,𝑋
Allowed substitution hint:   𝑆(𝑓)

Proof of Theorem fsetfocdm
Dummy variables 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsetfocdm.f . . . 4 𝐹 = {𝑓𝑓:𝐴𝐵}
2 fsetfocdm.s . . . 4 𝑆 = (𝑔𝐹 ↦ (𝑔𝑋))
31, 2fsetfcdm 8756 . . 3 (𝑋𝐴𝑆:𝐹𝐵)
43adantl 482 . 2 ((𝐴𝑉𝑋𝐴) → 𝑆:𝐹𝐵)
5 simplr 767 . . . . . 6 ((((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) ∧ 𝑥𝐴) → 𝑔𝐵)
6 eqid 2737 . . . . . 6 (𝑥𝐴𝑔) = (𝑥𝐴𝑔)
75, 6fmptd 7058 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔):𝐴𝐵)
8 simpll 765 . . . . . . 7 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → 𝐴𝑉)
98mptexd 7170 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔) ∈ V)
10 feq1 6646 . . . . . . 7 (𝑓 = (𝑥𝐴𝑔) → (𝑓:𝐴𝐵 ↔ (𝑥𝐴𝑔):𝐴𝐵))
1110, 1elab2g 3630 . . . . . 6 ((𝑥𝐴𝑔) ∈ V → ((𝑥𝐴𝑔) ∈ 𝐹 ↔ (𝑥𝐴𝑔):𝐴𝐵))
129, 11syl 17 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔) ∈ 𝐹 ↔ (𝑥𝐴𝑔):𝐴𝐵))
137, 12mpbird 256 . . . 4 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔) ∈ 𝐹)
14 fveq2 6839 . . . . . 6 ( = (𝑥𝐴𝑔) → (𝑆) = (𝑆‘(𝑥𝐴𝑔)))
1514eqeq2d 2748 . . . . 5 ( = (𝑥𝐴𝑔) → (𝑔 = (𝑆) ↔ 𝑔 = (𝑆‘(𝑥𝐴𝑔))))
1615adantl 482 . . . 4 ((((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) ∧ = (𝑥𝐴𝑔)) → (𝑔 = (𝑆) ↔ 𝑔 = (𝑆‘(𝑥𝐴𝑔))))
17 fveq1 6838 . . . . . . . . 9 (𝑔 = 𝑓 → (𝑔𝑋) = (𝑓𝑋))
1817cbvmptv 5216 . . . . . . . 8 (𝑔𝐹 ↦ (𝑔𝑋)) = (𝑓𝐹 ↦ (𝑓𝑋))
192, 18eqtri 2765 . . . . . . 7 𝑆 = (𝑓𝐹 ↦ (𝑓𝑋))
2019a1i 11 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → 𝑆 = (𝑓𝐹 ↦ (𝑓𝑋)))
21 fveq1 6838 . . . . . . 7 (𝑓 = (𝑥𝐴𝑔) → (𝑓𝑋) = ((𝑥𝐴𝑔)‘𝑋))
2221adantl 482 . . . . . 6 ((((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) ∧ 𝑓 = (𝑥𝐴𝑔)) → (𝑓𝑋) = ((𝑥𝐴𝑔)‘𝑋))
23 fvexd 6854 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔)‘𝑋) ∈ V)
2420, 22, 13, 23fvmptd 6952 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑆‘(𝑥𝐴𝑔)) = ((𝑥𝐴𝑔)‘𝑋))
25 eqidd 2738 . . . . . . 7 ((𝐴𝑉𝑋𝐴) → (𝑥𝐴𝑔) = (𝑥𝐴𝑔))
26 eqidd 2738 . . . . . . 7 (((𝐴𝑉𝑋𝐴) ∧ 𝑥 = 𝑋) → 𝑔 = 𝑔)
27 simpr 485 . . . . . . 7 ((𝐴𝑉𝑋𝐴) → 𝑋𝐴)
28 vex 3447 . . . . . . . 8 𝑔 ∈ V
2928a1i 11 . . . . . . 7 ((𝐴𝑉𝑋𝐴) → 𝑔 ∈ V)
3025, 26, 27, 29fvmptd 6952 . . . . . 6 ((𝐴𝑉𝑋𝐴) → ((𝑥𝐴𝑔)‘𝑋) = 𝑔)
3130adantr 481 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔)‘𝑋) = 𝑔)
3224, 31eqtr2d 2778 . . . 4 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → 𝑔 = (𝑆‘(𝑥𝐴𝑔)))
3313, 16, 32rspcedvd 3581 . . 3 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ∃𝐹 𝑔 = (𝑆))
3433ralrimiva 3141 . 2 ((𝐴𝑉𝑋𝐴) → ∀𝑔𝐵𝐹 𝑔 = (𝑆))
35 dffo3 7048 . 2 (𝑆:𝐹onto𝐵 ↔ (𝑆:𝐹𝐵 ∧ ∀𝑔𝐵𝐹 𝑔 = (𝑆)))
364, 34, 35sylanbrc 583 1 ((𝐴𝑉𝑋𝐴) → 𝑆:𝐹onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2714  wral 3062  wrex 3071  Vcvv 3443  cmpt 5186  wf 6489  ontowfo 6491  cfv 6493
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501
This theorem is referenced by:  fsetprcnex  8758
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