| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > f1oeq1d | Structured version Visualization version GIF version | ||
| Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| f1oeq1d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
| Ref | Expression |
|---|---|
| f1oeq1d | ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1oeq1d.1 | . 2 ⊢ (𝜑 → 𝐹 = 𝐺) | |
| 2 | f1oeq1 6836 | . 2 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 –1-1-onto→wf1o 6560 |
| 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-ext 2708 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 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-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 |
| This theorem is referenced by: f1orescnv 6863 f1osng 6889 f1ocoima 7323 f1ofvswap 7326 dif1en 9200 dif1enOLD 9202 cnfcomlem 9739 cnfcom2 9742 cnfcom3clem 9745 infxpenc 10058 infxpenc2lem2 10060 infxpenc2 10062 canthp1lem2 10693 pwfseqlem5 10703 pwfseq 10704 s2f1o 14955 s4f1o 14957 bitsf1ocnv 16481 yonffthlem 18327 grplactcnv 19061 eqgen 19199 znunithash 21583 tgpconncompeqg 24120 fcobijfs 32734 indf1o 32849 s2f1 32929 ccatws1f1o 32936 mgcf1o 32993 gsummpt2d 33052 gsumwrd2dccat 33070 subfacp1lem3 35187 subfacp1lem5 35189 ismrer1 37845 hvmap1o 41765 metakunt34 42239 3f1oss2 47088 |
| Copyright terms: Public domain | W3C validator |