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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 3766 | . . . . 5 ⊢ (𝑧 = 𝑦 → 〈𝐴, 𝑧〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | eleq1d 2239 | . . . 4 ⊢ (𝑧 = 𝑦 → (〈𝐴, 𝑧〉 ∈ 𝐹 ↔ 〈𝐴, 𝑦〉 ∈ 𝐹)) |
3 | tz6.12f.1 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 | |
4 | 3 | nfel2 2325 | . . . . . 6 ⊢ Ⅎ𝑦〈𝐴, 𝑧〉 ∈ 𝐹 |
5 | nfv 1521 | . . . . . 6 ⊢ Ⅎ𝑧〈𝐴, 𝑦〉 ∈ 𝐹 | |
6 | 4, 5, 2 | cbveu 2043 | . . . . 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 470 | . . 3 ⊢ (𝑧 = 𝑦 → ((〈𝐴, 𝑧〉 ∈ 𝐹 ∧ ∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹) ↔ (〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹))) |
9 | eqeq2 2180 | . . 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 5524 | . 2 ⊢ ((〈𝐴, 𝑧〉 ∈ 𝐹 ∧ ∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑧) | |
12 | 10, 11 | chvarv 1930 | 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 1348 ∃!weu 2019 ∈ wcel 2141 Ⅎwnfc 2299 〈cop 3586 ‘cfv 5198 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 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-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 |
This theorem is referenced by: (None) |
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