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| Mirrors > Home > ILE Home > Th. List > f1oeq23 | GIF version | ||
| Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.) |
| Ref | Expression |
|---|---|
| f1oeq23 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1oeq2 5511 | . 2 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶)) | |
| 2 | f1oeq3 5512 | . 2 ⊢ (𝐶 = 𝐷 → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷)) | |
| 3 | 1, 2 | sylan9bb 462 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1373 –1-1-onto→wf1o 5270 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-11 1529 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 |
| This theorem depends on definitions: df-bi 117 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-in 3172 df-ss 3179 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 |
| This theorem is referenced by: seqf1og 10666 zfz1isolem1 10985 |
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