![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > imadomg | Structured version Visualization version GIF version |
Description: An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.) |
Ref | Expression |
---|---|
imadomg | ⊢ (𝐴 ∈ 𝐵 → (Fun 𝐹 → (𝐹 “ 𝐴) ≼ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5326 | . . . 4 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
2 | resfunexg 6709 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ 𝐴) ∈ V) | |
3 | 2 | dmexd 7334 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → dom (𝐹 ↾ 𝐴) ∈ V) |
4 | funres 6144 | . . . . . . 7 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)) | |
5 | funforn 6339 | . . . . . . 7 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴)) | |
6 | 4, 5 | sylib 210 | . . . . . 6 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴)) |
7 | 6 | adantr 473 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴)) |
8 | fodomg 9634 | . . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ∈ V → ((𝐹 ↾ 𝐴):dom (𝐹 ↾ 𝐴)–onto→ran (𝐹 ↾ 𝐴) → ran (𝐹 ↾ 𝐴) ≼ dom (𝐹 ↾ 𝐴))) | |
9 | 3, 7, 8 | sylc 65 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → ran (𝐹 ↾ 𝐴) ≼ dom (𝐹 ↾ 𝐴)) |
10 | 1, 9 | syl5eqbr 4879 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ≼ dom (𝐹 ↾ 𝐴)) |
11 | 10 | expcom 403 | . 2 ⊢ (𝐴 ∈ 𝐵 → (Fun 𝐹 → (𝐹 “ 𝐴) ≼ dom (𝐹 ↾ 𝐴))) |
12 | dmres 5630 | . . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹) | |
13 | inss1 4029 | . . . . . 6 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴 | |
14 | 12, 13 | eqsstri 3832 | . . . . 5 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴 |
15 | ssdomg 8242 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → (dom (𝐹 ↾ 𝐴) ⊆ 𝐴 → dom (𝐹 ↾ 𝐴) ≼ 𝐴)) | |
16 | 14, 15 | mpi 20 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → dom (𝐹 ↾ 𝐴) ≼ 𝐴) |
17 | domtr 8249 | . . . 4 ⊢ (((𝐹 “ 𝐴) ≼ dom (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) ≼ 𝐴) → (𝐹 “ 𝐴) ≼ 𝐴) | |
18 | 16, 17 | sylan2 587 | . . 3 ⊢ (((𝐹 “ 𝐴) ≼ dom (𝐹 ↾ 𝐴) ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ≼ 𝐴) |
19 | 18 | expcom 403 | . 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 “ 𝐴) ≼ dom (𝐹 ↾ 𝐴) → (𝐹 “ 𝐴) ≼ 𝐴)) |
20 | 11, 19 | syld 47 | 1 ⊢ (𝐴 ∈ 𝐵 → (Fun 𝐹 → (𝐹 “ 𝐴) ≼ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 Vcvv 3386 ∩ cin 3769 ⊆ wss 3770 class class class wbr 4844 dom cdm 5313 ran crn 5314 ↾ cres 5315 “ cima 5316 Fun wfun 6096 –onto→wfo 6100 ≼ cdom 8194 |
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-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-ac2 9574 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-se 5273 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-isom 6111 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-er 7983 df-map 8098 df-en 8197 df-dom 8198 df-card 9052 df-acn 9055 df-ac 9226 |
This theorem is referenced by: fimact 9646 uniimadom 9655 hausmapdom 21631 |
Copyright terms: Public domain | W3C validator |