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Mirrors > Home > MPE Home > Th. List > tz6.12i | Structured version Visualization version GIF version |
Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
tz6.12i | ⊢ (𝐵 ≠ ∅ → ((𝐹‘𝐴) = 𝐵 → 𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6676 | . . . . 5 ⊢ (𝐹‘𝐴) ∈ V | |
2 | neeq1 3075 | . . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → ((𝐹‘𝐴) ≠ ∅ ↔ 𝑦 ≠ ∅)) | |
3 | tz6.12-2 6653 | . . . . . . . . . . 11 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅) | |
4 | 3 | necon1ai 3040 | . . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ≠ ∅ → ∃!𝑦 𝐴𝐹𝑦) |
5 | tz6.12c 6688 | . . . . . . . . . 10 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) | |
6 | 4, 5 | syl 17 | . . . . . . . . 9 ⊢ ((𝐹‘𝐴) ≠ ∅ → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
7 | 6 | biimpcd 250 | . . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹𝑦)) |
8 | 2, 7 | sylbird 261 | . . . . . . 7 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝑦 ≠ ∅ → 𝐴𝐹𝑦)) |
9 | 8 | eqcoms 2826 | . . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝑦 ≠ ∅ → 𝐴𝐹𝑦)) |
10 | neeq1 3075 | . . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝑦 ≠ ∅ ↔ (𝐹‘𝐴) ≠ ∅)) | |
11 | breq2 5061 | . . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝐴𝐹𝑦 ↔ 𝐴𝐹(𝐹‘𝐴))) | |
12 | 9, 10, 11 | 3imtr3d 294 | . . . . 5 ⊢ (𝑦 = (𝐹‘𝐴) → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴))) |
13 | 1, 12 | vtocle 3581 | . . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴)) |
14 | 13 | a1i 11 | . . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴))) |
15 | neeq1 3075 | . . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → ((𝐹‘𝐴) ≠ ∅ ↔ 𝐵 ≠ ∅)) | |
16 | breq2 5061 | . . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → (𝐴𝐹(𝐹‘𝐴) ↔ 𝐴𝐹𝐵)) | |
17 | 14, 15, 16 | 3imtr3d 294 | . 2 ⊢ ((𝐹‘𝐴) = 𝐵 → (𝐵 ≠ ∅ → 𝐴𝐹𝐵)) |
18 | 17 | com12 32 | 1 ⊢ (𝐵 ≠ ∅ → ((𝐹‘𝐴) = 𝐵 → 𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∃!weu 2646 ≠ wne 3013 ∅c0 4288 class class class wbr 5057 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 |
This theorem is referenced by: fvbr0 6690 fvclss 6992 dcomex 9857 |
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