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Mirrors > Home > ILE Home > Th. List > fvimacnv | GIF version |
Description: The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5316 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.) |
Ref | Expression |
---|---|
fvimacnv | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfvop 5652 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹) | |
2 | funfvex 5554 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ V) | |
3 | opelcnvg 4828 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹 ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹)) | |
4 | 2, 3 | sylancom 420 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹 ↔ 〈𝐴, (𝐹‘𝐴)〉 ∈ 𝐹)) |
5 | 1, 4 | mpbird 167 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹) |
6 | elimasng 5017 | . . . . 5 ⊢ (((𝐹‘𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) ↔ 〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹)) | |
7 | 2, 6 | sylancom 420 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) ↔ 〈(𝐹‘𝐴), 𝐴〉 ∈ ◡𝐹)) |
8 | 5, 7 | mpbird 167 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)})) |
9 | snssg 3744 | . . . . . . . 8 ⊢ ((𝐹‘𝐴) ∈ V → ((𝐹‘𝐴) ∈ 𝐵 ↔ {(𝐹‘𝐴)} ⊆ 𝐵)) | |
10 | 2, 9 | syl 14 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ {(𝐹‘𝐴)} ⊆ 𝐵)) |
11 | imass2 5025 | . . . . . . 7 ⊢ ({(𝐹‘𝐴)} ⊆ 𝐵 → (◡𝐹 “ {(𝐹‘𝐴)}) ⊆ (◡𝐹 “ 𝐵)) | |
12 | 10, 11 | biimtrdi 163 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 → (◡𝐹 “ {(𝐹‘𝐴)}) ⊆ (◡𝐹 “ 𝐵))) |
13 | 12 | imp 124 | . . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) ∧ (𝐹‘𝐴) ∈ 𝐵) → (◡𝐹 “ {(𝐹‘𝐴)}) ⊆ (◡𝐹 “ 𝐵)) |
14 | 13 | sseld 3169 | . . . 4 ⊢ (((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) ∧ (𝐹‘𝐴) ∈ 𝐵) → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) → 𝐴 ∈ (◡𝐹 “ 𝐵))) |
15 | 14 | ex 115 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) → 𝐴 ∈ (◡𝐹 “ 𝐵)))) |
16 | 8, 15 | mpid 42 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵))) |
17 | fvimacnvi 5654 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵) | |
18 | 17 | ex 115 | . . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵)) |
19 | 18 | adantr 276 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵)) |
20 | 16, 19 | impbid 129 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 {csn 3610 〈cop 3613 ◡ccnv 4646 dom cdm 4647 “ cima 4650 Fun wfun 5232 ‘cfv 5238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-fv 5246 |
This theorem is referenced by: funimass3 5656 elpreima 5659 fisumss 11441 psrbaglesuppg 13975 |
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