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Theorem f1ocnvfvrneq 5777
Description: If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
Assertion
Ref Expression
f1ocnvfvrneq ((𝐹:𝐴1-1𝐵 ∧ (𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))

Proof of Theorem f1ocnvfvrneq
StepHypRef Expression
1 f1f1orn 5468 . . 3 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
2 f1ocnv 5470 . . 3 (𝐹:𝐴1-1-onto→ran 𝐹𝐹:ran 𝐹1-1-onto𝐴)
3 f1of1 5456 . . 3 (𝐹:ran 𝐹1-1-onto𝐴𝐹:ran 𝐹1-1𝐴)
4 f1veqaeq 5764 . . . 4 ((𝐹:ran 𝐹1-1𝐴 ∧ (𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
54ex 115 . . 3 (𝐹:ran 𝐹1-1𝐴 → ((𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
61, 2, 3, 54syl 18 . 2 (𝐹:𝐴1-1𝐵 → ((𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷)))
76imp 124 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  ccnv 4622  ran crn 4624  1-1wf1 5209  1-1-ontowf1o 5211  cfv 5212
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 4118  ax-pow 4171  ax-pr 4206
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 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220
This theorem is referenced by: (None)
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