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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 4806 | . . . . 5 ⊢ (𝑧 = 𝑦 → 〈𝐴, 𝑧〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | eleq1d 2899 | . . . 4 ⊢ (𝑧 = 𝑦 → (〈𝐴, 𝑧〉 ∈ 𝐹 ↔ 〈𝐴, 𝑦〉 ∈ 𝐹)) |
3 | tz6.12f.1 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 | |
4 | 3 | nfel2 2998 | . . . . . 6 ⊢ Ⅎ𝑦〈𝐴, 𝑧〉 ∈ 𝐹 |
5 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑧〈𝐴, 𝑦〉 ∈ 𝐹 | |
6 | 4, 5, 2 | cbveuw 2690 | . . . . 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 2835 | . . 3 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝐴) = 𝑧 ↔ (𝐹‘𝐴) = 𝑦)) | |
10 | 8, 9 | imbi12d 347 | . 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 6695 | . 2 ⊢ ((〈𝐴, 𝑧〉 ∈ 𝐹 ∧ ∃!𝑧〈𝐴, 𝑧〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑧) | |
12 | 10, 11 | chvarvv 2005 | 1 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃!weu 2653 Ⅎwnfc 2963 〈cop 4575 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 |
This theorem is referenced by: (None) |
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