Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1elima GIF version

Theorem f1elima 5412
 Description: Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
f1elima ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ 𝑋𝑌))

Proof of Theorem f1elima
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 f1fn 5093 . . . 4 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
2 fvelimab 5229 . . . 4 ((𝐹 Fn 𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
31, 2sylan 267 . . 3 ((𝐹:𝐴1-1𝐵𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
433adant2 923 . 2 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
5 ssel 2939 . . . . . . . 8 (𝑌𝐴 → (𝑧𝑌𝑧𝐴))
65impac 363 . . . . . . 7 ((𝑌𝐴𝑧𝑌) → (𝑧𝐴𝑧𝑌))
7 f1fveq 5411 . . . . . . . . . . . 12 ((𝐹:𝐴1-1𝐵 ∧ (𝑧𝐴𝑋𝐴)) → ((𝐹𝑧) = (𝐹𝑋) ↔ 𝑧 = 𝑋))
87ancom2s 500 . . . . . . . . . . 11 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴𝑧𝐴)) → ((𝐹𝑧) = (𝐹𝑋) ↔ 𝑧 = 𝑋))
98biimpd 132 . . . . . . . . . 10 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴𝑧𝐴)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑧 = 𝑋))
109anassrs 380 . . . . . . . . 9 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑧𝐴) → ((𝐹𝑧) = (𝐹𝑋) → 𝑧 = 𝑋))
11 eleq1 2100 . . . . . . . . . 10 (𝑧 = 𝑋 → (𝑧𝑌𝑋𝑌))
1211biimpcd 148 . . . . . . . . 9 (𝑧𝑌 → (𝑧 = 𝑋𝑋𝑌))
1310, 12sylan9 389 . . . . . . . 8 ((((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑧𝐴) ∧ 𝑧𝑌) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1413anasss 379 . . . . . . 7 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ (𝑧𝐴𝑧𝑌)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
156, 14sylan2 270 . . . . . 6 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ (𝑌𝐴𝑧𝑌)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1615anassrs 380 . . . . 5 ((((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑌𝐴) ∧ 𝑧𝑌) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1716rexlimdva 2433 . . . 4 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
18173impa 1099 . . 3 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
19 eqid 2040 . . . 4 (𝐹𝑋) = (𝐹𝑋)
20 fveq2 5178 . . . . . 6 (𝑧 = 𝑋 → (𝐹𝑧) = (𝐹𝑋))
2120eqeq1d 2048 . . . . 5 (𝑧 = 𝑋 → ((𝐹𝑧) = (𝐹𝑋) ↔ (𝐹𝑋) = (𝐹𝑋)))
2221rspcev 2656 . . . 4 ((𝑋𝑌 ∧ (𝐹𝑋) = (𝐹𝑋)) → ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋))
2319, 22mpan2 401 . . 3 (𝑋𝑌 → ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋))
2418, 23impbid1 130 . 2 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) ↔ 𝑋𝑌))
254, 24bitrd 177 1 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ 𝑋𝑌))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∃wrex 2307   ⊆ wss 2917   “ cima 4348   Fn wfn 4897  –1-1→wf1 4899  ‘cfv 4902 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fv 4910 This theorem is referenced by:  f1imass  5413
 Copyright terms: Public domain W3C validator