![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-fvimacnv0 | Structured version Visualization version GIF version |
Description: Variant of fvimacnv 6800 where membership of 𝐴 in the domain is not needed provided the containing class 𝐵 does not contain the empty set. Note that this antecedent would not be needed with definition df-afv 43676. (Contributed by BJ, 7-Jan-2024.) |
Ref | Expression |
---|---|
bj-fvimacnv0 | ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2877 | . . . . . . . . . 10 ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹‘𝐴) ∈ 𝐵 ↔ ∅ ∈ 𝐵)) | |
2 | 1 | biimpcd 252 | . . . . . . . . 9 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → ((𝐹‘𝐴) = ∅ → ∅ ∈ 𝐵)) |
3 | 2 | con3rr3 158 | . . . . . . . 8 ⊢ (¬ ∅ ∈ 𝐵 → ((𝐹‘𝐴) ∈ 𝐵 → ¬ (𝐹‘𝐴) = ∅)) |
4 | 3 | imp 410 | . . . . . . 7 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → ¬ (𝐹‘𝐴) = ∅) |
5 | ndmfv 6675 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
6 | 4, 5 | nsyl2 143 | . . . . . 6 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → 𝐴 ∈ dom 𝐹) |
7 | simpr 488 | . . . . . 6 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → (𝐹‘𝐴) ∈ 𝐵) | |
8 | fvimacnv 6800 | . . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) | |
9 | 8 | biimpd 232 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵))) |
10 | 9 | ex 416 | . . . . . . 7 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵)))) |
11 | 10 | com3l 89 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝐵 → (Fun 𝐹 → 𝐴 ∈ (◡𝐹 “ 𝐵)))) |
12 | 6, 7, 11 | sylc 65 | . . . . 5 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → (Fun 𝐹 → 𝐴 ∈ (◡𝐹 “ 𝐵))) |
13 | 12 | ex 416 | . . . 4 ⊢ (¬ ∅ ∈ 𝐵 → ((𝐹‘𝐴) ∈ 𝐵 → (Fun 𝐹 → 𝐴 ∈ (◡𝐹 “ 𝐵)))) |
14 | 13 | com3r 87 | . . 3 ⊢ (Fun 𝐹 → (¬ ∅ ∈ 𝐵 → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵)))) |
15 | 14 | imp 410 | . 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵))) |
16 | fvimacnvi 6799 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵) | |
17 | 16 | ex 416 | . . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵)) |
18 | 17 | adantr 484 | . 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵)) |
19 | 15, 18 | impbid 215 | 1 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∅c0 4243 ◡ccnv 5518 dom cdm 5519 “ cima 5522 Fun wfun 6318 ‘cfv 6324 |
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 |
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-ne 2988 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-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-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: bj-isrvec 34708 |
Copyright terms: Public domain | W3C validator |