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Theorem foelrn 5730
Description: Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof shortened by BJ, 6-Jul-2022.)
Assertion
Ref Expression
foelrn ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem foelrn
StepHypRef Expression
1 foima2 5729 . 2 (𝐹:𝐴onto𝐵 → (𝐶𝐵 ↔ ∃𝑥𝐴 𝐶 = (𝐹𝑥)))
21biimpa 294 1 ((𝐹:𝐴onto𝐵𝐶𝐵) → ∃𝑥𝐴 𝐶 = (𝐹𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  wrex 2449  ontowfo 5194  cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fo 5202  df-fv 5204
This theorem is referenced by:  foco2  5731  ctmlemr  7083  ctm  7084  ctssdclemn0  7085  ctssdccl  7086  ctssdc  7088  enumctlemm  7089  fodju0  7121  exmidfodomrlemr  7172  exmidfodomrlemrALT  7173  ennnfonelemrn  12367  ctinf  12378  ctiunctlemfo  12387  subctctexmid  13999  pw1nct  14001
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