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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 6334 | . . 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 6335 | . . 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 6336 | . . 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 297 | 1 ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1658 –1-1→wf1 6121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 |
This theorem is referenced by: f10d 6412 fthf1 16930 cofth 16948 istrkgld 25772 istrkg2ld 25773 isushgr 26360 isuspgr 26452 isusgr 26453 isuspgrop 26461 isusgrop 26462 ausgrusgrb 26465 ausgrusgri 26468 usgrstrrepe 26533 uspgr1e 26542 usgrexmpl 26561 usgrres1 26613 usgrexi 26740 uspgr2wlkeq 26944 usgr2trlncl 27063 aciunf1 30013 |
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