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Theorem fsetfocdm 8900
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 8899 . . 3 (𝑋𝐴𝑆:𝐹𝐵)
43adantl 481 . 2 ((𝐴𝑉𝑋𝐴) → 𝑆:𝐹𝐵)
5 simplr 769 . . . . . 6 ((((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) ∧ 𝑥𝐴) → 𝑔𝐵)
65fmpttd 7135 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔):𝐴𝐵)
7 simpll 767 . . . . . . 7 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → 𝐴𝑉)
87mptexd 7244 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔) ∈ V)
9 feq1 6717 . . . . . . 7 (𝑓 = (𝑥𝐴𝑔) → (𝑓:𝐴𝐵 ↔ (𝑥𝐴𝑔):𝐴𝐵))
109, 1elab2g 3683 . . . . . 6 ((𝑥𝐴𝑔) ∈ V → ((𝑥𝐴𝑔) ∈ 𝐹 ↔ (𝑥𝐴𝑔):𝐴𝐵))
118, 10syl 17 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔) ∈ 𝐹 ↔ (𝑥𝐴𝑔):𝐴𝐵))
126, 11mpbird 257 . . . 4 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔) ∈ 𝐹)
13 fveq1 6906 . . . . . . . . 9 (𝑔 = 𝑓 → (𝑔𝑋) = (𝑓𝑋))
1413cbvmptv 5261 . . . . . . . 8 (𝑔𝐹 ↦ (𝑔𝑋)) = (𝑓𝐹 ↦ (𝑓𝑋))
152, 14eqtri 2763 . . . . . . 7 𝑆 = (𝑓𝐹 ↦ (𝑓𝑋))
16 fveq1 6906 . . . . . . 7 (𝑓 = (𝑥𝐴𝑔) → (𝑓𝑋) = ((𝑥𝐴𝑔)‘𝑋))
17 fvexd 6922 . . . . . . 7 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔)‘𝑋) ∈ V)
1815, 16, 12, 17fvmptd3 7039 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑆‘(𝑥𝐴𝑔)) = ((𝑥𝐴𝑔)‘𝑋))
19 eqidd 2736 . . . . . . . 8 (𝑥 = 𝑋𝑔 = 𝑔)
20 eqid 2735 . . . . . . . 8 (𝑥𝐴𝑔) = (𝑥𝐴𝑔)
21 vex 3482 . . . . . . . 8 𝑔 ∈ V
2219, 20, 21fvmpt 7016 . . . . . . 7 (𝑋𝐴 → ((𝑥𝐴𝑔)‘𝑋) = 𝑔)
2322ad2antlr 727 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔)‘𝑋) = 𝑔)
2418, 23eqtrd 2775 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑆‘(𝑥𝐴𝑔)) = 𝑔)
25 fveq2 6907 . . . . . 6 ( = (𝑥𝐴𝑔) → (𝑆) = (𝑆‘(𝑥𝐴𝑔)))
2625eqcomd 2741 . . . . 5 ( = (𝑥𝐴𝑔) → (𝑆‘(𝑥𝐴𝑔)) = (𝑆))
2724, 26sylan9req 2796 . . . 4 ((((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) ∧ = (𝑥𝐴𝑔)) → 𝑔 = (𝑆))
2812, 27rspcedeq2vd 3630 . . 3 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ∃𝐹 𝑔 = (𝑆))
2928ralrimiva 3144 . 2 ((𝐴𝑉𝑋𝐴) → ∀𝑔𝐵𝐹 𝑔 = (𝑆))
30 dffo3 7122 . 2 (𝑆:𝐹onto𝐵 ↔ (𝑆:𝐹𝐵 ∧ ∀𝑔𝐵𝐹 𝑔 = (𝑆)))
314, 29, 30sylanbrc 583 1 ((𝐴𝑉𝑋𝐴) → 𝑆:𝐹onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cmpt 5231  wf 6559  ontowfo 6561  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  fsetprcnex  8901
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