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Mirrors > Home > MPE Home > Th. List > Mathboxes > tz6.12i-afv2 | Structured version Visualization version GIF version |
Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6671. (Contributed by AV, 5-Sep-2022.) |
Ref | Expression |
---|---|
tz6.12i-afv2 | ⊢ (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2877 | . . . . . . . . 9 ⊢ ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝑦 ∈ ran 𝐹)) | |
2 | dfatafv2rnb 43783 | . . . . . . . . . . . 12 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹) | |
3 | dfdfat2 43684 | . . . . . . . . . . . . 13 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦)) | |
4 | 3 | simprbi 500 | . . . . . . . . . . . 12 ⊢ (𝐹 defAt 𝐴 → ∃!𝑦 𝐴𝐹𝑦) |
5 | 2, 4 | sylbir 238 | . . . . . . . . . . 11 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ∃!𝑦 𝐴𝐹𝑦) |
6 | tz6.12c-afv2 43798 | . . . . . . . . . . 11 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) | |
7 | 5, 6 | syl 17 | . . . . . . . . . 10 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
8 | 7 | biimpcd 252 | . . . . . . . . 9 ⊢ ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹𝑦)) |
9 | 1, 8 | sylbird 263 | . . . . . . . 8 ⊢ ((𝐹''''𝐴) = 𝑦 → (𝑦 ∈ ran 𝐹 → 𝐴𝐹𝑦)) |
10 | 9 | eqcoms 2806 | . . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 → 𝐴𝐹𝑦)) |
11 | eleq1 2877 | . . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹)) | |
12 | breq2 5034 | . . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝐴𝐹𝑦 ↔ 𝐴𝐹(𝐹''''𝐴))) | |
13 | 10, 11, 12 | 3imtr3d 296 | . . . . . 6 ⊢ (𝑦 = (𝐹''''𝐴) → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹(𝐹''''𝐴))) |
14 | 13 | vtocleg 3529 | . . . . 5 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹(𝐹''''𝐴))) |
15 | 14 | pm2.43i 52 | . . . 4 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹(𝐹''''𝐴)) |
16 | 15 | a1i 11 | . . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹(𝐹''''𝐴))) |
17 | eleq1 2877 | . . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝐵 ∈ ran 𝐹)) | |
18 | breq2 5034 | . . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵)) | |
19 | 16, 17, 18 | 3imtr3d 296 | . 2 ⊢ ((𝐹''''𝐴) = 𝐵 → (𝐵 ∈ ran 𝐹 → 𝐴𝐹𝐵)) |
20 | 19 | com12 32 | 1 ⊢ (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃!weu 2628 class class class wbr 5030 dom cdm 5519 ran crn 5520 defAt wdfat 43672 ''''cafv2 43764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fn 6327 df-dfat 43675 df-afv2 43765 |
This theorem is referenced by: (None) |
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