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| Mirrors > Home > MPE Home > Th. List > isoeq4 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
| Ref | Expression |
|---|---|
| isoeq4 | ⊢ (𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1oeq2 6837 | . . 3 ⊢ (𝐴 = 𝐶 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻:𝐶–1-1-onto→𝐵)) | |
| 2 | raleq 3323 | . . . 4 ⊢ (𝐴 = 𝐶 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) | |
| 3 | 2 | raleqbi1dv 3338 | . . 3 ⊢ (𝐴 = 𝐶 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) |
| 4 | 1, 3 | anbi12d 632 | . 2 ⊢ (𝐴 = 𝐶 → ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐶–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))) |
| 5 | df-isom 6570 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) | |
| 6 | df-isom 6570 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ (𝐻:𝐶–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) | |
| 7 | 4, 5, 6 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∀wral 3061 class class class wbr 5143 –1-1-onto→wf1o 6560 ‘cfv 6561 Isom wiso 6562 |
| 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-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ral 3062 df-rex 3071 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-isom 6570 |
| This theorem is referenced by: oieu 9579 oiid 9581 finnisoeu 10153 iunfictbso 10154 fz1isolem 14500 isercolllem3 15703 summolem2a 15751 prodmolem2a 15970 erdszelem1 35196 erdsze 35207 erdsze2lem1 35208 erdsze2lem2 35209 isoeq145d 43432 fzisoeu 45312 fourierdlem36 46158 fourierdlem112 46233 fourierdlem113 46234 |
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