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Theorem f1elima 5777
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 5425 . . . 4 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
2 fvelimab 5575 . . . 4 ((𝐹 Fn 𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
31, 2sylan 283 . . 3 ((𝐹:𝐴1-1𝐵𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
433adant2 1016 . 2 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋)))
5 ssel 3151 . . . . . . . 8 (𝑌𝐴 → (𝑧𝑌𝑧𝐴))
65impac 381 . . . . . . 7 ((𝑌𝐴𝑧𝑌) → (𝑧𝐴𝑧𝑌))
7 f1fveq 5776 . . . . . . . . . . . 12 ((𝐹:𝐴1-1𝐵 ∧ (𝑧𝐴𝑋𝐴)) → ((𝐹𝑧) = (𝐹𝑋) ↔ 𝑧 = 𝑋))
87ancom2s 566 . . . . . . . . . . 11 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴𝑧𝐴)) → ((𝐹𝑧) = (𝐹𝑋) ↔ 𝑧 = 𝑋))
98biimpd 144 . . . . . . . . . 10 ((𝐹:𝐴1-1𝐵 ∧ (𝑋𝐴𝑧𝐴)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑧 = 𝑋))
109anassrs 400 . . . . . . . . 9 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑧𝐴) → ((𝐹𝑧) = (𝐹𝑋) → 𝑧 = 𝑋))
11 eleq1 2240 . . . . . . . . . 10 (𝑧 = 𝑋 → (𝑧𝑌𝑋𝑌))
1211biimpcd 159 . . . . . . . . 9 (𝑧𝑌 → (𝑧 = 𝑋𝑋𝑌))
1310, 12sylan9 409 . . . . . . . 8 ((((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑧𝐴) ∧ 𝑧𝑌) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1413anasss 399 . . . . . . 7 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ (𝑧𝐴𝑧𝑌)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
156, 14sylan2 286 . . . . . 6 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ (𝑌𝐴𝑧𝑌)) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1615anassrs 400 . . . . 5 ((((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑌𝐴) ∧ 𝑧𝑌) → ((𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
1716rexlimdva 2594 . . . 4 (((𝐹:𝐴1-1𝐵𝑋𝐴) ∧ 𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
18173impa 1194 . . 3 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) → 𝑋𝑌))
19 eqid 2177 . . . 4 (𝐹𝑋) = (𝐹𝑋)
20 fveq2 5517 . . . . . 6 (𝑧 = 𝑋 → (𝐹𝑧) = (𝐹𝑋))
2120eqeq1d 2186 . . . . 5 (𝑧 = 𝑋 → ((𝐹𝑧) = (𝐹𝑋) ↔ (𝐹𝑋) = (𝐹𝑋)))
2221rspcev 2843 . . . 4 ((𝑋𝑌 ∧ (𝐹𝑋) = (𝐹𝑋)) → ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋))
2319, 22mpan2 425 . . 3 (𝑋𝑌 → ∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋))
2418, 23impbid1 142 . 2 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → (∃𝑧𝑌 (𝐹𝑧) = (𝐹𝑋) ↔ 𝑋𝑌))
254, 24bitrd 188 1 ((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ 𝑋𝑌))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  wrex 2456  wss 3131  cima 4631   Fn wfn 5213  1-1wf1 5215  cfv 5218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fv 5226
This theorem is referenced by:  f1imass  5778  iseqf1olemnab  10491  fprodssdc  11601  ctinfom  12432  ssnnctlemct  12450
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