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Mirrors > Home > ILE Home > Th. List > elpreima | GIF version |
Description: Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
elpreima | ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ (◡𝐹 “ 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvimass 4949 | . . . . 5 ⊢ (◡𝐹 “ 𝐶) ⊆ dom 𝐹 | |
2 | 1 | sseli 3124 | . . . 4 ⊢ (𝐵 ∈ (◡𝐹 “ 𝐶) → 𝐵 ∈ dom 𝐹) |
3 | fndm 5269 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
4 | 3 | eleq2d 2227 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴)) |
5 | 2, 4 | syl5ib 153 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ (◡𝐹 “ 𝐶) → 𝐵 ∈ 𝐴)) |
6 | fnfun 5267 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
7 | fvimacnvi 5581 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ (◡𝐹 “ 𝐶)) → (𝐹‘𝐵) ∈ 𝐶) | |
8 | 6, 7 | sylan 281 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ (◡𝐹 “ 𝐶)) → (𝐹‘𝐵) ∈ 𝐶) |
9 | 8 | ex 114 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ (◡𝐹 “ 𝐶) → (𝐹‘𝐵) ∈ 𝐶)) |
10 | 5, 9 | jcad 305 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ (◡𝐹 “ 𝐶) → (𝐵 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ 𝐶))) |
11 | fvimacnv 5582 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ((𝐹‘𝐵) ∈ 𝐶 ↔ 𝐵 ∈ (◡𝐹 “ 𝐶))) | |
12 | 11 | funfni 5270 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) ∈ 𝐶 ↔ 𝐵 ∈ (◡𝐹 “ 𝐶))) |
13 | 12 | biimpd 143 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) ∈ 𝐶 → 𝐵 ∈ (◡𝐹 “ 𝐶))) |
14 | 13 | expimpd 361 | . 2 ⊢ (𝐹 Fn 𝐴 → ((𝐵 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ 𝐶) → 𝐵 ∈ (◡𝐹 “ 𝐶))) |
15 | 10, 14 | impbid 128 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ (◡𝐹 “ 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2128 ◡ccnv 4585 dom cdm 4586 “ cima 4589 Fun wfun 5164 Fn wfn 5165 ‘cfv 5170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-fv 5178 |
This theorem is referenced by: fniniseg 5587 fncnvima2 5588 rexsupp 5591 unpreima 5592 respreima 5595 fisumss 11289 fprodssdc 11487 tanvalap 11605 cncnpi 12639 cncnp 12641 cnpdis 12653 tx1cn 12680 tx2cn 12681 txcnmpt 12684 txdis1cn 12689 xmeterval 12846 cnbl0 12945 cnblcld 12946 |
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