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Mirrors > Home > ILE Home > Th. List > f1eq1 | GIF version |
Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
f1eq1 | ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq1 5225 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) | |
2 | cnveq 4683 | . . . 4 ⊢ (𝐹 = 𝐺 → ◡𝐹 = ◡𝐺) | |
3 | 2 | funeqd 5115 | . . 3 ⊢ (𝐹 = 𝐺 → (Fun ◡𝐹 ↔ Fun ◡𝐺)) |
4 | 1, 3 | anbi12d 464 | . 2 ⊢ (𝐹 = 𝐺 → ((𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺))) |
5 | df-f1 5098 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
6 | df-f1 5098 | . 2 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) | |
7 | 4, 5, 6 | 3bitr4g 222 | 1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ◡ccnv 4508 Fun wfun 5087 ⟶wf 5089 –1-1→wf1 5090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 |
This theorem is referenced by: f1oeq1 5326 f1eq123d 5330 fun11iun 5356 fo00 5371 tposf12 6134 f1dom2g 6618 f1domg 6620 dom3d 6636 domtr 6647 djudom 6946 difinfsn 6953 djudoml 7043 djudomr 7044 |
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