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| Mirrors > Home > MPE Home > Th. List > f1fveq | Structured version Visualization version GIF version | ||
| Description: Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.) |
| Ref | Expression |
|---|---|
| f1fveq | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) ↔ 𝐶 = 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1veqaeq 7277 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷)) | |
| 2 | fveq2 6906 | . 2 ⊢ (𝐶 = 𝐷 → (𝐹‘𝐶) = (𝐹‘𝐷)) | |
| 3 | 1, 2 | impbid1 225 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) ↔ 𝐶 = 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 –1-1→wf1 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-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-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fv 6569 |
| This theorem is referenced by: f1elima 7283 f1dom3fv3dif 7288 cocan1 7311 isof1oidb 7344 isosolem 7367 f1oiso 7371 weniso 7374 f1oweALT 7997 2dom 9070 xpdom2 9107 wemapwe 9737 fseqenlem1 10064 dfac12lem2 10185 infpssrlem4 10346 fin23lem28 10380 isf32lem7 10399 iundom2g 10580 canthnumlem 10688 canthwelem 10690 canthp1lem2 10693 pwfseqlem4 10702 seqf1olem1 14082 bitsinv2 16480 bitsf1 16483 sadasslem 16507 sadeq 16509 bitsuz 16511 eulerthlem2 16819 f1ocpbllem 17569 f1ovscpbl 17571 fthi 17965 f1omvdmvd 19461 odf1 19580 dprdf1o 20052 zntoslem 21575 iporthcom 21653 ply1scln0 22295 cnt0 23354 cnhaus 23362 imasdsf1olem 24383 imasf1oxmet 24385 dyadmbl 25635 vitalilem3 25645 dvcnvlem 26014 facth1 26206 usgredg2v 29244 mndlactf1o 33035 mndractf1o 33036 cycpmco2lem6 33151 erdszelem9 35204 cvmliftmolem1 35286 msubff1 35561 metf1o 37762 rngoisocnv 37988 laut11 40088 aks6d1c6lem3 42173 gicabl 43111 fourierdlem50 46171 isuspgrim0lem 47871 |
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