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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 5367 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) | |
2 | cnveq 4819 | . . . 4 ⊢ (𝐹 = 𝐺 → ◡𝐹 = ◡𝐺) | |
3 | 2 | funeqd 5257 | . . 3 ⊢ (𝐹 = 𝐺 → (Fun ◡𝐹 ↔ Fun ◡𝐺)) |
4 | 1, 3 | anbi12d 473 | . 2 ⊢ (𝐹 = 𝐺 → ((𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺))) |
5 | df-f1 5240 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
6 | df-f1 5240 | . 2 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) | |
7 | 4, 5, 6 | 3bitr4g 223 | 1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ◡ccnv 4643 Fun wfun 5229 ⟶wf 5231 –1-1→wf1 5232 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 |
This theorem is referenced by: f1oeq1 5468 f1eq123d 5472 fun11iun 5501 fo00 5516 tposf12 6295 f1dom2g 6783 f1domg 6785 dom3d 6801 domtr 6812 djudom 7123 difinfsn 7130 djudoml 7249 djudomr 7250 4sqlem11 12436 nninfdc 12507 conjsubgen 13234 |
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