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Theorem f1elima 7265
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 6788 . . . 4 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
2 fvelimab 6964 . . . 4 ((𝐹 Fn 𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
31, 2sylan 579 . . 3 ((𝐹:𝐴1-1𝐵𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
433adant2 1130 . 2 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
5 ssel 3975 . . . . . . . 8 (𝑌𝐴 → (𝑧𝑌𝑧𝐴))
65impac 552 . . . . . . 7 ((𝑌𝐴𝑧𝑌) → (𝑧𝐴𝑧𝑌))
7 f1fveq 7264 . . . . . . . . . . . 12 ((𝐹:𝐴1-1𝐵 ∧ (𝑧𝐴𝑋𝐴)) → ((𝐹𝑧) = (𝐹𝑋) ↔ 𝑧 = 𝑋))
87ancom2s 647 . . . . . . . . . . 11 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴𝑧𝐴)) → ((𝐹𝑧) = (𝐹𝑋) ↔ 𝑧 = 𝑋))
98biimpd 228 . . . . . . . . . 10 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴𝑧𝐴)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑧 = 𝑋))
109anassrs 467 . . . . . . . . 9 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑧𝐴) → ((𝐹𝑧) = (𝐹𝑋) → 𝑧 = 𝑋))
11 eleq1 2820 . . . . . . . . . 10 (𝑧 = 𝑋 → (𝑧𝑌𝑋𝑌))
1211biimpcd 248 . . . . . . . . 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 592 . . . . . 6 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ (𝑌𝐴𝑧𝑌)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1615anassrs 467 . . . . 5 ((((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑌𝐴) ∧ 𝑧𝑌) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1716rexlimdva 3154 . . . 4 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
18173impa 1109 . . 3 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
19 eqid 2731 . . . 4 (𝐹𝑋) = (𝐹𝑋)
20 fveqeq2 6900 . . . . 5 (𝑧 = 𝑋 → ((𝐹𝑧) = (𝐹𝑋) ↔ (𝐹𝑋) = (𝐹𝑋)))
2120rspcev 3612 . . . 4 ((𝑋𝑌 ∧ (𝐹𝑋) = (𝐹𝑋)) → ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋))
2219, 21mpan2 688 . . 3 (𝑋𝑌 → ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋))
2318, 22impbid1 224 . 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 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wrex 3069  wss 3948  cima 5679   Fn wfn 6538  1-1wf1 6540  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551
This theorem is referenced by:  f1imass  7266  domunfican  9326  acndom2  10055  hashf1lem1  14422  hashf1lem1OLD  14423  f1omvdconj  19362  gsumzaddlem  19837  lindfmm  21692  axcontlem10  28665  trlsegvdeg  29914  ismtyima  37137
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