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

Theorem foima2 5544
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 5251). (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 5251 . . . 4 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
21eqcomd 2094 . . 3 (𝐹:𝐴onto𝐵𝐵 = (𝐹𝐴))
32eleq2d 2158 . 2 (𝐹:𝐴onto𝐵 → (𝑌𝐵𝑌 ∈ (𝐹𝐴)))
4 fofn 5248 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
5 ssid 3045 . . 3 𝐴𝐴
6 fvelimab 5373 . . . 4 ((𝐹 Fn 𝐴𝐴𝐴) → (𝑌 ∈ (𝐹𝐴) ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑌))
7 eqcom 2091 . . . . 5 ((𝐹𝑥) = 𝑌𝑌 = (𝐹𝑥))
87rexbii 2386 . . . 4 (∃𝑥𝐴 (𝐹𝑥) = 𝑌 ↔ ∃𝑥𝐴 𝑌 = (𝐹𝑥))
96, 8syl6bb 195 . . 3 ((𝐹 Fn 𝐴𝐴𝐴) → (𝑌 ∈ (𝐹𝐴) ↔ ∃𝑥𝐴 𝑌 = (𝐹𝑥)))
104, 5, 9sylancl 405 . 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 1290  wcel 1439  wrex 2361  wss 3000  cima 4455   Fn wfn 5023  ontowfo 5026  cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fo 5034  df-fv 5036
This theorem is referenced by:  foelrn  5545
  Copyright terms: Public domain W3C validator