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Mirrors > Home > MPE Home > Th. List > rnct | Structured version Visualization version GIF version |
Description: The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
Ref | Expression |
---|---|
rnct | ⊢ (𝐴 ≼ ω → ran 𝐴 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvct 8194 | . 2 ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω) | |
2 | dmct 9534 | . 2 ⊢ (◡𝐴 ≼ ω → dom ◡𝐴 ≼ ω) | |
3 | df-rn 5273 | . . . 4 ⊢ ran 𝐴 = dom ◡𝐴 | |
4 | 3 | breq1i 4807 | . . 3 ⊢ (ran 𝐴 ≼ ω ↔ dom ◡𝐴 ≼ ω) |
5 | 4 | biimpri 218 | . 2 ⊢ (dom ◡𝐴 ≼ ω → ran 𝐴 ≼ ω) |
6 | 1, 2, 5 | 3syl 18 | 1 ⊢ (𝐴 ≼ ω → ran 𝐴 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 4800 ◡ccnv 5261 dom cdm 5262 ran crn 5263 ωcom 7226 ≼ cdom 8115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-rep 4919 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-ac2 9473 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-ral 3051 df-rex 3052 df-reu 3053 df-rmo 3054 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-pss 3727 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4585 df-int 4624 df-iun 4670 df-br 4801 df-opab 4861 df-mpt 4878 df-tr 4901 df-id 5170 df-eprel 5175 df-po 5183 df-so 5184 df-fr 5221 df-se 5222 df-we 5223 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-pred 5837 df-ord 5883 df-on 5884 df-suc 5886 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-isom 6054 df-riota 6770 df-ov 6812 df-oprab 6813 df-mpt2 6814 df-1st 7329 df-2nd 7330 df-wrecs 7572 df-recs 7633 df-er 7907 df-map 8021 df-en 8118 df-dom 8119 df-card 8951 df-acn 8954 df-ac 9125 |
This theorem is referenced by: abrexctf 29801 sigapildsys 30530 dya2iocct 30647 omssubadd 30667 carsgclctunlem2 30686 pmeasadd 30692 smfpimcc 41516 |
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