![]() |
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 5428 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵)) | |
2 | foeq1 5446 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵)) | |
3 | 1, 2 | anbi12d 473 | . 2 ⊢ (𝐹 = 𝐺 → ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ (𝐺:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–onto→𝐵))) |
4 | df-f1o 5235 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | |
5 | df-f1o 5235 | . 2 ⊢ (𝐺:𝐴–1-1-onto→𝐵 ↔ (𝐺:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–onto→𝐵)) | |
6 | 3, 4, 5 | 3bitr4g 223 | 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 104 ↔ wb 105 = wceq 1363 –1-1→wf1 5225 –onto→wfo 5226 –1-1-onto→wf1o 5227 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-br 4016 df-opab 4077 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 |
This theorem is referenced by: f1oeq123d 5467 f1oeq1d 5468 f1ocnvb 5487 f1orescnv 5489 f1ovi 5512 f1osng 5514 f1oresrab 5694 fsn 5701 isoeq1 5815 mapsn 6704 mapsnf1o3 6711 f1oen3g 6768 ensn1 6810 xpcomf1o 6839 xpen 6859 seq3f1olemstep 10515 seq3f1olemp 10516 fihasheqf1oi 10781 fihashf1rn 10782 hashfacen 10830 summodc 11405 fsum3 11409 prodmodc 11600 fprodseq 11605 eulerthlemh 12245 relogf1o 14578 |
Copyright terms: Public domain | W3C validator |