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Theorem f1elima 7283
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 6805 . . . 4 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
2 fvelimab 6981 . . . 4 ((𝐹 Fn 𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
31, 2sylan 580 . . 3 ((𝐹:𝐴1-1𝐵𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
433adant2 1132 . 2 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
5 ssel 3977 . . . . . . . 8 (𝑌𝐴 → (𝑧𝑌𝑧𝐴))
65impac 552 . . . . . . 7 ((𝑌𝐴𝑧𝑌) → (𝑧𝐴𝑧𝑌))
7 f1fveq 7282 . . . . . . . . . . . 12 ((𝐹:𝐴1-1𝐵 ∧ (𝑧𝐴𝑋𝐴)) → ((𝐹𝑧) = (𝐹𝑋) ↔ 𝑧 = 𝑋))
87ancom2s 650 . . . . . . . . . . 11 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴𝑧𝐴)) → ((𝐹𝑧) = (𝐹𝑋) ↔ 𝑧 = 𝑋))
98biimpd 229 . . . . . . . . . 10 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴𝑧𝐴)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑧 = 𝑋))
109anassrs 467 . . . . . . . . 9 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑧𝐴) → ((𝐹𝑧) = (𝐹𝑋) → 𝑧 = 𝑋))
11 eleq1 2829 . . . . . . . . . 10 (𝑧 = 𝑋 → (𝑧𝑌𝑋𝑌))
1211biimpcd 249 . . . . . . . . 9 (𝑧𝑌 → (𝑧 = 𝑋𝑋𝑌))
1310, 12sylan9 507 . . . . . . . 8 ((((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑧𝐴) ∧ 𝑧𝑌) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1413anasss 466 . . . . . . 7 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ (𝑧𝐴𝑧𝑌)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
156, 14sylan2 593 . . . . . 6 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ (𝑌𝐴𝑧𝑌)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1615anassrs 467 . . . . 5 ((((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑌𝐴) ∧ 𝑧𝑌) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1716rexlimdva 3155 . . . 4 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
18173impa 1110 . . 3 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
19 eqid 2737 . . . 4 (𝐹𝑋) = (𝐹𝑋)
20 fveqeq2 6915 . . . . 5 (𝑧 = 𝑋 → ((𝐹𝑧) = (𝐹𝑋) ↔ (𝐹𝑋) = (𝐹𝑋)))
2120rspcev 3622 . . . 4 ((𝑋𝑌 ∧ (𝐹𝑋) = (𝐹𝑋)) → ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋))
2219, 21mpan2 691 . . 3 (𝑋𝑌 → ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋))
2318, 22impbid1 225 . 2 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) ↔ 𝑋𝑌))
244, 23bitrd 279 1 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ 𝑋𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  wss 3951  cima 5688   Fn wfn 6556  1-1wf1 6558  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569
This theorem is referenced by:  f1imass  7284  domunfican  9361  acndom2  10094  hashf1lem1  14494  f1omvdconj  19464  gsumzaddlem  19939  lindfmm  21847  axcontlem10  28988  trlsegvdeg  30246  ismtyima  37810
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