![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > abrexct | Structured version Visualization version GIF version |
Description: An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
Ref | Expression |
---|---|
abrexct | ⊢ (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2772 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | rnmpt 5663 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
3 | 1stcrestlem 21754 | . 2 ⊢ (𝐴 ≼ ω → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) | |
4 | 2, 3 | syl5eqbrr 4959 | 1 ⊢ (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 {cab 2753 ∃wrex 3083 class class class wbr 4923 ↦ cmpt 5002 ran crn 5401 ωcom 7390 ≼ cdom 8296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-card 9154 df-acn 9157 |
This theorem is referenced by: sigaclcu2 30981 |
Copyright terms: Public domain | W3C validator |