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Theorem foima2 5621
Description: Given an onto function, an element is in its codomain if and only if it is the image of an element of its domain (see foima 5320). (Contributed by BJ, 6-Jul-2022.)
Assertion
Ref Expression
foima2 (𝐹:𝐴onto𝐵 → (𝑌𝐵 ↔ ∃𝑥𝐴 𝑌 = (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑌   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem foima2
StepHypRef Expression
1 foima 5320 . . . 4 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
21eqcomd 2123 . . 3 (𝐹:𝐴onto𝐵𝐵 = (𝐹𝐴))
32eleq2d 2187 . 2 (𝐹:𝐴onto𝐵 → (𝑌𝐵𝑌 ∈ (𝐹𝐴)))
4 fofn 5317 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
5 ssid 3087 . . 3 𝐴𝐴
6 fvelimab 5445 . . . 4 ((𝐹 Fn 𝐴𝐴𝐴) → (𝑌 ∈ (𝐹𝐴) ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
7 eqcom 2119 . . . . 5 ((𝐹𝑥) = 𝑌𝑌 = (𝐹𝑥))
87rexbii 2419 . . . 4 (∃𝑥𝐴 (𝐹𝑥) = 𝑌 ↔ ∃𝑥𝐴 𝑌 = (𝐹𝑥))
96, 8syl6bb 195 . . 3 ((𝐹 Fn 𝐴𝐴𝐴) → (𝑌 ∈ (𝐹𝐴) ↔ ∃𝑥𝐴 𝑌 = (𝐹𝑥)))
104, 5, 9sylancl 409 . 2 (𝐹:𝐴onto𝐵 → (𝑌 ∈ (𝐹𝐴) ↔ ∃𝑥𝐴 𝑌 = (𝐹𝑥)))
113, 10bitrd 187 1 (𝐹:𝐴onto𝐵 → (𝑌𝐵 ↔ ∃𝑥𝐴 𝑌 = (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  wrex 2394  wss 3041  cima 4512   Fn wfn 5088  ontowfo 5091  cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101
This theorem is referenced by:  foelrn  5622
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