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Theorem fsetfocdm 8551
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 8550 . . 3 (𝑋𝐴𝑆:𝐹𝐵)
43adantl 485 . 2 ((𝐴𝑉𝑋𝐴) → 𝑆:𝐹𝐵)
5 simplr 769 . . . . . 6 ((((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) ∧ 𝑥𝐴) → 𝑔𝐵)
6 eqid 2738 . . . . . 6 (𝑥𝐴𝑔) = (𝑥𝐴𝑔)
75, 6fmptd 6940 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔):𝐴𝐵)
8 simpll 767 . . . . . . 7 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → 𝐴𝑉)
98mptexd 7049 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔) ∈ V)
10 feq1 6535 . . . . . . 7 (𝑓 = (𝑥𝐴𝑔) → (𝑓:𝐴𝐵 ↔ (𝑥𝐴𝑔):𝐴𝐵))
1110, 1elab2g 3596 . . . . . 6 ((𝑥𝐴𝑔) ∈ V → ((𝑥𝐴𝑔) ∈ 𝐹 ↔ (𝑥𝐴𝑔):𝐴𝐵))
129, 11syl 17 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔) ∈ 𝐹 ↔ (𝑥𝐴𝑔):𝐴𝐵))
137, 12mpbird 260 . . . 4 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔) ∈ 𝐹)
14 fveq2 6726 . . . . . 6 ( = (𝑥𝐴𝑔) → (𝑆) = (𝑆‘(𝑥𝐴𝑔)))
1514eqeq2d 2749 . . . . 5 ( = (𝑥𝐴𝑔) → (𝑔 = (𝑆) ↔ 𝑔 = (𝑆‘(𝑥𝐴𝑔))))
1615adantl 485 . . . 4 ((((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) ∧ = (𝑥𝐴𝑔)) → (𝑔 = (𝑆) ↔ 𝑔 = (𝑆‘(𝑥𝐴𝑔))))
17 fveq1 6725 . . . . . . . . 9 (𝑔 = 𝑓 → (𝑔𝑋) = (𝑓𝑋))
1817cbvmptv 5167 . . . . . . . 8 (𝑔𝐹 ↦ (𝑔𝑋)) = (𝑓𝐹 ↦ (𝑓𝑋))
192, 18eqtri 2766 . . . . . . 7 𝑆 = (𝑓𝐹 ↦ (𝑓𝑋))
2019a1i 11 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → 𝑆 = (𝑓𝐹 ↦ (𝑓𝑋)))
21 fveq1 6725 . . . . . . 7 (𝑓 = (𝑥𝐴𝑔) → (𝑓𝑋) = ((𝑥𝐴𝑔)‘𝑋))
2221adantl 485 . . . . . 6 ((((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) ∧ 𝑓 = (𝑥𝐴𝑔)) → (𝑓𝑋) = ((𝑥𝐴𝑔)‘𝑋))
23 fvexd 6741 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔)‘𝑋) ∈ V)
2420, 22, 13, 23fvmptd 6834 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑆‘(𝑥𝐴𝑔)) = ((𝑥𝐴𝑔)‘𝑋))
25 eqidd 2739 . . . . . . 7 ((𝐴𝑉𝑋𝐴) → (𝑥𝐴𝑔) = (𝑥𝐴𝑔))
26 eqidd 2739 . . . . . . 7 (((𝐴𝑉𝑋𝐴) ∧ 𝑥 = 𝑋) → 𝑔 = 𝑔)
27 simpr 488 . . . . . . 7 ((𝐴𝑉𝑋𝐴) → 𝑋𝐴)
28 vex 3419 . . . . . . . 8 𝑔 ∈ V
2928a1i 11 . . . . . . 7 ((𝐴𝑉𝑋𝐴) → 𝑔 ∈ V)
3025, 26, 27, 29fvmptd 6834 . . . . . 6 ((𝐴𝑉𝑋𝐴) → ((𝑥𝐴𝑔)‘𝑋) = 𝑔)
3130adantr 484 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔)‘𝑋) = 𝑔)
3224, 31eqtr2d 2779 . . . 4 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → 𝑔 = (𝑆‘(𝑥𝐴𝑔)))
3313, 16, 32rspcedvd 3547 . . 3 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ∃𝐹 𝑔 = (𝑆))
3433ralrimiva 3106 . 2 ((𝐴𝑉𝑋𝐴) → ∀𝑔𝐵𝐹 𝑔 = (𝑆))
35 dffo3 6930 . 2 (𝑆:𝐹onto𝐵 ↔ (𝑆:𝐹𝐵 ∧ ∀𝑔𝐵𝐹 𝑔 = (𝑆)))
364, 34, 35sylanbrc 586 1 ((𝐴𝑉𝑋𝐴) → 𝑆:𝐹onto𝐵)
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  cmpt 5144  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:  fsetprcnex  8552
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