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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 5694 | . . 3 ⊢ (◡𝐹 “ 𝐵) ⊆ ran ◡𝐹 | |
2 | dfdm4 5519 | . . . 4 ⊢ dom 𝐹 = ran ◡𝐹 | |
3 | fdm 6264 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
4 | ssid 3819 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
5 | 3, 4 | syl6eqss 3851 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴) |
6 | 2, 5 | syl5eqssr 3846 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran ◡𝐹 ⊆ 𝐴) |
7 | 1, 6 | syl5ss 3809 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) ⊆ 𝐴) |
8 | imassrn 5694 | . . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹 | |
9 | frn 6262 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
10 | 8, 9 | syl5ss 3809 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵) |
11 | ffun 6259 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
12 | 4, 3 | syl5sseqr 3850 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹) |
13 | funimass3 6559 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) | |
14 | 11, 12, 13 | syl2anc 580 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) |
15 | 10, 14 | mpbid 224 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ (◡𝐹 “ 𝐵)) |
16 | 7, 15 | eqssd 3815 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ⊆ wss 3769 ◡ccnv 5311 dom cdm 5312 ran crn 5313 “ cima 5315 Fun wfun 6095 ⟶wf 6097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-fv 6109 |
This theorem is referenced by: fimacnvinrn 6574 fmpt 6606 frnsuppeq 7544 fin1a2lem7 9516 cnclima 21401 iscncl 21402 cnindis 21425 cncmp 21524 ptrescn 21771 qtopuni 21834 qtopcld 21845 qtopcmap 21851 ordthmeolem 21933 rnelfmlem 22084 mbfdm 23734 ismbf 23736 mbfimaicc 23739 ismbf2d 23748 ismbf3d 23762 mbfimaopn2 23765 i1fd 23789 plyeq0 24308 fsumcvg4 30512 zrhunitpreima 30538 imambfm 30840 carsggect 30896 dstrvprob 31050 poimirlem30 33928 dvtan 33948 smfresal 41741 |
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