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Theorem f1elima 6421
 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 6015 . . . 4 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
2 fvelimab 6163 . . . 4 ((𝐹 Fn 𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
31, 2sylan 487 . . 3 ((𝐹:𝐴1-1𝐵𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
433adant2 1073 . 2 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
5 ssel 3562 . . . . . . . 8 (𝑌𝐴 → (𝑧𝑌𝑧𝐴))
65impac 649 . . . . . . 7 ((𝑌𝐴𝑧𝑌) → (𝑧𝐴𝑧𝑌))
7 f1fveq 6420 . . . . . . . . . . . 12 ((𝐹:𝐴1-1𝐵 ∧ (𝑧𝐴𝑋𝐴)) → ((𝐹𝑧) = (𝐹𝑋) ↔ 𝑧 = 𝑋))
87ancom2s 840 . . . . . . . . . . 11 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴𝑧𝐴)) → ((𝐹𝑧) = (𝐹𝑋) ↔ 𝑧 = 𝑋))
98biimpd 218 . . . . . . . . . 10 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴𝑧𝐴)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑧 = 𝑋))
109anassrs 678 . . . . . . . . 9 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑧𝐴) → ((𝐹𝑧) = (𝐹𝑋) → 𝑧 = 𝑋))
11 eleq1 2676 . . . . . . . . . 10 (𝑧 = 𝑋 → (𝑧𝑌𝑋𝑌))
1211biimpcd 238 . . . . . . . . 9 (𝑧𝑌 → (𝑧 = 𝑋𝑋𝑌))
1310, 12sylan9 687 . . . . . . . 8 ((((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑧𝐴) ∧ 𝑧𝑌) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1413anasss 677 . . . . . . 7 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ (𝑧𝐴𝑧𝑌)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
156, 14sylan2 490 . . . . . 6 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ (𝑌𝐴𝑧𝑌)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1615anassrs 678 . . . . 5 ((((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑌𝐴) ∧ 𝑧𝑌) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1716rexlimdva 3013 . . . 4 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
18173impa 1251 . . 3 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
19 eqid 2610 . . . 4 (𝐹𝑋) = (𝐹𝑋)
20 fveq2 6103 . . . . . 6 (𝑧 = 𝑋 → (𝐹𝑧) = (𝐹𝑋))
2120eqeq1d 2612 . . . . 5 (𝑧 = 𝑋 → ((𝐹𝑧) = (𝐹𝑋) ↔ (𝐹𝑋) = (𝐹𝑋)))
2221rspcev 3282 . . . 4 ((𝑋𝑌 ∧ (𝐹𝑋) = (𝐹𝑋)) → ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋))
2319, 22mpan2 703 . . 3 (𝑋𝑌 → ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋))
2418, 23impbid1 214 . 2 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) ↔ 𝑋𝑌))
254, 24bitrd 267 1 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ 𝑋𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540   “ cima 5041   Fn wfn 5799  –1-1→wf1 5801  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812 This theorem is referenced by:  f1imass  6422  domunfican  8118  acndom2  8760  hashf1lem1  13096  f1omvdconj  17689  gsumzaddlem  18144  lindfmm  19985  axcontlem10  25653  eupath2lem3  26506  ismtyima  32772  trlsegvdeg  41395
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