Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > f1oeq1 | GIF version |
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
f1oeq1 | ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq1 5323 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵)) | |
2 | foeq1 5341 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵)) | |
3 | 1, 2 | anbi12d 464 | . 2 ⊢ (𝐹 = 𝐺 → ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ (𝐺:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–onto→𝐵))) |
4 | df-f1o 5130 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | |
5 | df-f1o 5130 | . 2 ⊢ (𝐺:𝐴–1-1-onto→𝐵 ↔ (𝐺:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–onto→𝐵)) | |
6 | 3, 4, 5 | 3bitr4g 222 | 1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 –1-1→wf1 5120 –onto→wfo 5121 –1-1-onto→wf1o 5122 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 |
This theorem is referenced by: f1oeq123d 5362 f1ocnvb 5381 f1orescnv 5383 f1ovi 5406 f1osng 5408 f1oresrab 5585 fsn 5592 isoeq1 5702 mapsn 6584 mapsnf1o3 6591 f1oen3g 6648 ensn1 6690 xpcomf1o 6719 xpen 6739 seq3f1olemstep 10274 seq3f1olemp 10275 fihasheqf1oi 10534 fihashf1rn 10535 hashfacen 10579 summodc 11152 fsum3 11156 prodmodc 11347 |
Copyright terms: Public domain | W3C validator |