![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > f1eq123d | Structured version Visualization version GIF version |
Description: Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
f1eq123d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
f1eq123d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
f1eq123d.3 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
f1eq123d | ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq123d.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | f1eq1 6544 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐴–1-1→𝐶)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐴–1-1→𝐶)) |
4 | f1eq123d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
5 | f1eq2 6545 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐺:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐶)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐺:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐶)) |
7 | f1eq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
8 | f1eq3 6546 | . . 3 ⊢ (𝐶 = 𝐷 → (𝐺:𝐵–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷)) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → (𝐺:𝐵–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷)) |
10 | 3, 6, 9 | 3bitrd 308 | 1 ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 –1-1→wf1 6321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 |
This theorem is referenced by: f10d 6623 fthf1 17179 cofth 17197 istrkgld 26253 istrkg2ld 26254 isushgr 26854 isuspgr 26945 isusgr 26946 isuspgrop 26954 isusgrop 26955 ausgrusgrb 26958 ausgrusgri 26961 usgrstrrepe 27025 uspgr1e 27034 usgrexmpl 27053 usgrres1 27105 usgrexi 27231 uspgr2wlkeq 27435 usgr2trlncl 27549 aciunf1 30426 pfxf1 30644 s1f1 30645 tocycfv 30801 tocycf 30809 tocyc01 30810 cycpmco2f1 30816 cycpmco2rn 30817 cycpmco2lem1 30818 cycpmco2lem2 30819 cycpmco2lem3 30820 cycpmco2lem4 30821 cycpmco2lem5 30822 cycpmco2lem6 30823 cycpmco2lem7 30824 cycpmco2 30825 cycpm3cl2 30828 cycpmconjv 30834 tocyccntz 30836 cyc3evpm 30842 cycpmgcl 30845 cycpmconjslem2 30847 cyc3conja 30849 dimkerim 31111 f1resfz0f1d 32471 fundcmpsurinjALT 43929 |
Copyright terms: Public domain | W3C validator |