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Theorem fsetfocdm 8857
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 8856 . . 3 (𝑋𝐴𝑆:𝐹𝐵)
43adantl 481 . 2 ((𝐴𝑉𝑋𝐴) → 𝑆:𝐹𝐵)
5 simplr 766 . . . . . 6 ((((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) ∧ 𝑥𝐴) → 𝑔𝐵)
6 eqid 2726 . . . . . 6 (𝑥𝐴𝑔) = (𝑥𝐴𝑔)
75, 6fmptd 7109 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔):𝐴𝐵)
8 simpll 764 . . . . . . 7 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → 𝐴𝑉)
98mptexd 7221 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔) ∈ V)
10 feq1 6692 . . . . . . 7 (𝑓 = (𝑥𝐴𝑔) → (𝑓:𝐴𝐵 ↔ (𝑥𝐴𝑔):𝐴𝐵))
1110, 1elab2g 3665 . . . . . 6 ((𝑥𝐴𝑔) ∈ V → ((𝑥𝐴𝑔) ∈ 𝐹 ↔ (𝑥𝐴𝑔):𝐴𝐵))
129, 11syl 17 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔) ∈ 𝐹 ↔ (𝑥𝐴𝑔):𝐴𝐵))
137, 12mpbird 257 . . . 4 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑥𝐴𝑔) ∈ 𝐹)
14 fveq2 6885 . . . . . 6 ( = (𝑥𝐴𝑔) → (𝑆) = (𝑆‘(𝑥𝐴𝑔)))
1514eqeq2d 2737 . . . . 5 ( = (𝑥𝐴𝑔) → (𝑔 = (𝑆) ↔ 𝑔 = (𝑆‘(𝑥𝐴𝑔))))
1615adantl 481 . . . 4 ((((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) ∧ = (𝑥𝐴𝑔)) → (𝑔 = (𝑆) ↔ 𝑔 = (𝑆‘(𝑥𝐴𝑔))))
17 fveq1 6884 . . . . . . . . 9 (𝑔 = 𝑓 → (𝑔𝑋) = (𝑓𝑋))
1817cbvmptv 5254 . . . . . . . 8 (𝑔𝐹 ↦ (𝑔𝑋)) = (𝑓𝐹 ↦ (𝑓𝑋))
192, 18eqtri 2754 . . . . . . 7 𝑆 = (𝑓𝐹 ↦ (𝑓𝑋))
2019a1i 11 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → 𝑆 = (𝑓𝐹 ↦ (𝑓𝑋)))
21 fveq1 6884 . . . . . . 7 (𝑓 = (𝑥𝐴𝑔) → (𝑓𝑋) = ((𝑥𝐴𝑔)‘𝑋))
2221adantl 481 . . . . . 6 ((((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) ∧ 𝑓 = (𝑥𝐴𝑔)) → (𝑓𝑋) = ((𝑥𝐴𝑔)‘𝑋))
23 fvexd 6900 . . . . . 6 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔)‘𝑋) ∈ V)
2420, 22, 13, 23fvmptd 6999 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → (𝑆‘(𝑥𝐴𝑔)) = ((𝑥𝐴𝑔)‘𝑋))
25 eqidd 2727 . . . . . . 7 ((𝐴𝑉𝑋𝐴) → (𝑥𝐴𝑔) = (𝑥𝐴𝑔))
26 eqidd 2727 . . . . . . 7 (((𝐴𝑉𝑋𝐴) ∧ 𝑥 = 𝑋) → 𝑔 = 𝑔)
27 simpr 484 . . . . . . 7 ((𝐴𝑉𝑋𝐴) → 𝑋𝐴)
28 vex 3472 . . . . . . . 8 𝑔 ∈ V
2928a1i 11 . . . . . . 7 ((𝐴𝑉𝑋𝐴) → 𝑔 ∈ V)
3025, 26, 27, 29fvmptd 6999 . . . . . 6 ((𝐴𝑉𝑋𝐴) → ((𝑥𝐴𝑔)‘𝑋) = 𝑔)
3130adantr 480 . . . . 5 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ((𝑥𝐴𝑔)‘𝑋) = 𝑔)
3224, 31eqtr2d 2767 . . . 4 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → 𝑔 = (𝑆‘(𝑥𝐴𝑔)))
3313, 16, 32rspcedvd 3608 . . 3 (((𝐴𝑉𝑋𝐴) ∧ 𝑔𝐵) → ∃𝐹 𝑔 = (𝑆))
3433ralrimiva 3140 . 2 ((𝐴𝑉𝑋𝐴) → ∀𝑔𝐵𝐹 𝑔 = (𝑆))
35 dffo3 7097 . 2 (𝑆:𝐹onto𝐵 ↔ (𝑆:𝐹𝐵 ∧ ∀𝑔𝐵𝐹 𝑔 = (𝑆)))
364, 34, 35sylanbrc 582 1 ((𝐴𝑉𝑋𝐴) → 𝑆:𝐹onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  {cab 2703  wral 3055  wrex 3064  Vcvv 3468  cmpt 5224  wf 6533  ontowfo 6535  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545
This theorem is referenced by:  fsetprcnex  8858
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