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Theorem foelrnf 39687
Description: Property of a surjective function. As foelrn 6418 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
foelrnf.1 𝑥𝐹
Assertion
Ref Expression
foelrnf ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem foelrnf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 foelrnf.1 . . . 4 𝑥𝐹
21dffo3f 39678 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
32simprbi 479 . 2 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
4 eqeq1 2655 . . . 4 (𝑦 = 𝐶 → (𝑦 = (𝐹𝑥) ↔ 𝐶 = (𝐹𝑥)))
54rexbidv 3081 . . 3 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐴 𝐶 = (𝐹𝑥)))
65rspccva 3339 . 2 ((∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥) ∧ 𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
73, 6sylan 487 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wnfc 2780  wral 2941  wrex 2942  wf 5922  ontowfo 5924  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934
This theorem is referenced by:  fompt  39693
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