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Mirrors > Home > ILE Home > Th. List > tz6.12f | 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 3706 | . . . . 5 ⊢ (𝑧 = 𝑦 → 〈𝐴, 𝑧〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | eleq1d 2208 | . . . 4 ⊢ (𝑧 = 𝑦 → (〈𝐴, 𝑧〉 ∈ 𝐹 ↔ 〈𝐴, 𝑦〉 ∈ 𝐹)) |
3 | tz6.12f.1 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 | |
4 | 3 | nfel2 2294 | . . . . . 6 ⊢ Ⅎ𝑦〈𝐴, 𝑧〉 ∈ 𝐹 |
5 | nfv 1508 | . . . . . 6 ⊢ Ⅎ𝑧〈𝐴, 𝑦〉 ∈ 𝐹 | |
6 | 4, 5, 2 | cbveu 2023 | . . . . 5 ⊢ (∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹 ↔ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
7 | 6 | a1i 9 | . . . 4 ⊢ (𝑧 = 𝑦 → (∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹 ↔ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹)) |
8 | 2, 7 | anbi12d 464 | . . 3 ⊢ (𝑧 = 𝑦 → ((〈𝐴, 𝑧〉 ∈ 𝐹 ∧ ∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹) ↔ (〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹))) |
9 | eqeq2 2149 | . . 3 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝐴) = 𝑧 ↔ (𝐹‘𝐴) = 𝑦)) | |
10 | 8, 9 | imbi12d 233 | . 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 5449 | . 2 ⊢ ((〈𝐴, 𝑧〉 ∈ 𝐹 ∧ ∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑧) | |
12 | 10, 11 | chvarv 1909 | 1 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∃!weu 1999 Ⅎwnfc 2268 〈cop 3530 ‘cfv 5123 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 |
This theorem is referenced by: (None) |
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