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Mirrors > Home > MPE Home > Th. List > fimacnv | Structured version Visualization version 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 5618 | . . 3 ⊢ (◡𝐹 “ 𝐵) ⊆ ran ◡𝐹 | |
2 | dfdm4 5454 | . . . 4 ⊢ dom 𝐹 = ran ◡𝐹 | |
3 | fdm 6191 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
4 | ssid 3773 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
5 | 3, 4 | syl6eqss 3804 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴) |
6 | 2, 5 | syl5eqssr 3799 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran ◡𝐹 ⊆ 𝐴) |
7 | 1, 6 | syl5ss 3763 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) ⊆ 𝐴) |
8 | imassrn 5618 | . . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹 | |
9 | frn 6193 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
10 | 8, 9 | syl5ss 3763 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵) |
11 | ffun 6188 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
12 | 4, 3 | syl5sseqr 3803 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹) |
13 | funimass3 6476 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) | |
14 | 11, 12, 13 | syl2anc 565 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) |
15 | 10, 14 | mpbid 222 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ (◡𝐹 “ 𝐵)) |
16 | 7, 15 | eqssd 3769 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1631 ⊆ wss 3723 ◡ccnv 5248 dom cdm 5249 ran crn 5250 “ cima 5252 Fun wfun 6025 ⟶wf 6027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-fv 6039 |
This theorem is referenced by: fimacnvinrn 6491 fmpt 6523 frnsuppeq 7457 fin1a2lem7 9429 cnclima 21292 iscncl 21293 cnindis 21316 cncmp 21415 ptrescn 21662 qtopuni 21725 qtopcld 21736 qtopcmap 21742 ordthmeolem 21824 rnelfmlem 21975 mbfdm 23613 ismbf 23615 mbfimaicc 23618 ismbf2d 23627 ismbf3d 23640 mbfimaopn2 23643 i1fd 23667 plyeq0 24186 fsumcvg4 30333 zrhunitpreima 30359 imambfm 30661 carsggect 30717 dstrvprob 30870 poimirlem30 33768 dvtan 33788 smfresal 41511 |
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