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Mirrors > Home > ILE Home > Th. List > fimacnv | GIF version |
Description: The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.) |
Ref | Expression |
---|---|
fimacnv | ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imassrn 4982 | . . 3 ⊢ (◡𝐹 “ 𝐵) ⊆ ran ◡𝐹 | |
2 | dfdm4 4820 | . . . 4 ⊢ dom 𝐹 = ran ◡𝐹 | |
3 | fdm 5372 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
4 | ssid 3176 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
5 | 3, 4 | eqsstrdi 3208 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴) |
6 | 2, 5 | eqsstrrid 3203 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran ◡𝐹 ⊆ 𝐴) |
7 | 1, 6 | sstrid 3167 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) ⊆ 𝐴) |
8 | imassrn 4982 | . . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹 | |
9 | frn 5375 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
10 | 8, 9 | sstrid 3167 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵) |
11 | ffun 5369 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
12 | 4, 3 | sseqtrrid 3207 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹) |
13 | funimass3 5633 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) | |
14 | 11, 12, 13 | syl2anc 411 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) |
15 | 10, 14 | mpbid 147 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ (◡𝐹 “ 𝐵)) |
16 | 7, 15 | eqssd 3173 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ⊆ wss 3130 ◡ccnv 4626 dom cdm 4627 ran crn 4628 “ cima 4630 Fun wfun 5211 ⟶wf 5213 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2740 df-sbc 2964 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-fv 5225 |
This theorem is referenced by: fmpt 5667 nn0supp 9228 cnclima 13726 |
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