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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 4879 | . . . . 5 ⊢ (𝑧 = 𝑦 → 〈𝐴, 𝑧〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | eleq1d 2824 | . . . 4 ⊢ (𝑧 = 𝑦 → (〈𝐴, 𝑧〉 ∈ 𝐹 ↔ 〈𝐴, 𝑦〉 ∈ 𝐹)) |
3 | tz6.12f.1 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 | |
4 | 3 | nfel2 2922 | . . . . . 6 ⊢ Ⅎ𝑦〈𝐴, 𝑧〉 ∈ 𝐹 |
5 | nfv 1912 | . . . . . 6 ⊢ Ⅎ𝑧〈𝐴, 𝑦〉 ∈ 𝐹 | |
6 | 4, 5, 2 | cbveuw 2604 | . . . . 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 2747 | . . 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 6932 | . 2 ⊢ ((〈𝐴, 𝑧〉 ∈ 𝐹 ∧ ∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑧) | |
12 | 10, 11 | chvarvv 1996 | 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 1537 ∈ wcel 2106 ∃!weu 2566 Ⅎwnfc 2888 〈cop 4637 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 |
This theorem is referenced by: (None) |
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