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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 5432 | . 2 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶)) | |
2 | f1oeq3 5433 | . 2 ⊢ (𝐶 = 𝐷 → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷)) | |
3 | 1, 2 | sylan9bb 459 | 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 103 ↔ wb 104 = wceq 1348 –1-1-onto→wf1o 5197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-in 3127 df-ss 3134 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 |
This theorem is referenced by: zfz1isolem1 10775 |
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