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| Mirrors > Home > MPE Home > Th. List > tz6.12f | Structured version Visualization version GIF version | ||
| Description: Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.) | 
| Ref | Expression | 
|---|---|
| tz6.12f.1 | ⊢ Ⅎ𝑦𝐹 | 
| Ref | Expression | 
|---|---|
| tz6.12f | ⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opeq2 4873 | . . . . 5 ⊢ (𝑧 = 𝑦 → 〈𝐴, 𝑧〉 = 〈𝐴, 𝑦〉) | |
| 2 | 1 | eleq1d 2825 | . . . 4 ⊢ (𝑧 = 𝑦 → (〈𝐴, 𝑧〉 ∈ 𝐹 ↔ 〈𝐴, 𝑦〉 ∈ 𝐹)) | 
| 3 | tz6.12f.1 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 | |
| 4 | 3 | nfel2 2923 | . . . . . 6 ⊢ Ⅎ𝑦〈𝐴, 𝑧〉 ∈ 𝐹 | 
| 5 | nfv 1913 | . . . . . 6 ⊢ Ⅎ𝑧〈𝐴, 𝑦〉 ∈ 𝐹 | |
| 6 | 4, 5, 2 | cbveuw 2605 | . . . . 5 ⊢ (∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹 ↔ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) | 
| 7 | 6 | a1i 11 | . . . 4 ⊢ (𝑧 = 𝑦 → (∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹 ↔ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹)) | 
| 8 | 2, 7 | anbi12d 632 | . . 3 ⊢ (𝑧 = 𝑦 → ((〈𝐴, 𝑧〉 ∈ 𝐹 ∧ ∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹) ↔ (〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹))) | 
| 9 | eqeq2 2748 | . . 3 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝐴) = 𝑧 ↔ (𝐹‘𝐴) = 𝑦)) | |
| 10 | 8, 9 | imbi12d 344 | . 2 ⊢ (𝑧 = 𝑦 → (((〈𝐴, 𝑧〉 ∈ 𝐹 ∧ ∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑧) ↔ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦))) | 
| 11 | tz6.12 6930 | . 2 ⊢ ((〈𝐴, 𝑧〉 ∈ 𝐹 ∧ ∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑧) | |
| 12 | 10, 11 | chvarvv 1997 | 1 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃!weu 2567 Ⅎwnfc 2889 〈cop 4631 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: (None) | 
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