ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  foima2 GIF version

Theorem foima2 5646
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 5345). (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 5345 . . . 4 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
21eqcomd 2143 . . 3 (𝐹:𝐴onto𝐵𝐵 = (𝐹𝐴))
32eleq2d 2207 . 2 (𝐹:𝐴onto𝐵 → (𝑌𝐵𝑌 ∈ (𝐹𝐴)))
4 fofn 5342 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
5 ssid 3112 . . 3 𝐴𝐴
6 fvelimab 5470 . . . 4 ((𝐹 Fn 𝐴𝐴𝐴) → (𝑌 ∈ (𝐹𝐴) ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
7 eqcom 2139 . . . . 5 ((𝐹𝑥) = 𝑌𝑌 = (𝐹𝑥))
87rexbii 2440 . . . 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 1331  wcel 1480  wrex 2415  wss 3066  cima 4537   Fn wfn 5113  ontowfo 5116  cfv 5118
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126
This theorem is referenced by:  foelrn  5647
  Copyright terms: Public domain W3C validator