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Mirrors > Home > MPE Home > Th. List > f1oeq123d | Structured version Visualization version GIF version |
Description: Equality deduction for one-to-one onto 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 |
---|---|
f1oeq123d | ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq123d.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | f1oeq1 6380 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐴–1-1-onto→𝐶)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐴–1-1-onto→𝐶)) |
4 | f1eq123d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
5 | f1oeq2 6381 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐺:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐶)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐺:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐶)) |
7 | f1eq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
8 | f1oeq3 6382 | . . 3 ⊢ (𝐶 = 𝐷 → (𝐺:𝐵–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷)) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → (𝐺:𝐵–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷)) |
10 | 3, 6, 9 | 3bitrd 297 | 1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 –1-1-onto→wf1o 6134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 |
This theorem is referenced by: f1oprswap 6434 f1oprg 6435 f1ossf1o 6660 cnfcom 8894 ackbij2lem2 9397 s2f1o 14067 s4f1o 14069 idffth 16978 ressffth 16983 symg1bas 18199 symg2bas 18201 symgfixels 18237 symgfixelsi 18238 rhmf1o 19121 mat1f1o 20689 isismt 25885 ushgredgedg 26575 ushgredgedgloop 26577 ushgredgedgloopOLD 26578 trlreslem 27050 trlreslemOLD 27052 wlknwwlksnbij 27242 wwlksnextbij 27271 wwlksnextbijOLD 27272 clwlknf1oclwwlkn 27483 clwlknf1oclwwlknOLD 27485 eupth0 27617 eupthp1 27620 foresf1o 29905 f1ocnt 30123 dimkerim 30441 indf1ofs 30686 eulerpartgbij 31032 eulerpartlemn 31041 reprpmtf1o 31306 poimirlem16 34051 poimirlem17 34052 poimirlem19 34054 poimirlem20 34055 poimirlem28 34063 wessf1ornlem 40294 disjf1o 40301 ssnnf1octb 40305 sge0fodjrnlem 41557 f1oresf1orab 42330 isomgr 42736 rnghmf1o 42918 |
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