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| 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 6799 | . . 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 6800 | . . 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 6801 | . . 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 305 | 1 ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 –1-1→wf1 6558 |
| 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 |
| This theorem is referenced by: f10d 6882 fthf1 17964 cofth 17982 rngqiprngimf1 21310 istrkgld 28467 istrkg2ld 28468 isushgr 29078 isuspgr 29169 isusgr 29170 isuspgrop 29178 isusgrop 29179 ausgrusgrb 29182 ausgrusgri 29185 usgrstrrepe 29252 uspgr1e 29261 usgrres1 29332 usgrexi 29458 uspgr2wlkeq 29664 usgr2trlncl 29780 aciunf1 32673 pfxf1 32926 s1f1 32927 tocycfv 33129 tocycf 33137 tocyc01 33138 cycpmco2f1 33144 cycpmco2rn 33145 cycpmco2lem1 33146 cycpmco2lem2 33147 cycpmco2lem3 33148 cycpmco2lem4 33149 cycpmco2lem5 33150 cycpmco2lem6 33151 cycpmco2lem7 33152 cycpmco2 33153 cycpm3cl2 33156 cycpmconjv 33162 tocyccntz 33164 cyc3evpm 33170 cycpmgcl 33173 cycpmconjslem2 33175 cyc3conja 33177 dimkerim 33678 f1resfz0f1d 35119 aks6d1c2 42131 f1cof1b 47089 fundcmpsurinjALT 47399 stgrusgra 47926 gpgusgra 48012 |
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