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Mirrors > Home > MPE Home > Th. List > numdom | Structured version Visualization version GIF version |
Description: A set dominated by a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
numdom | ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardon 9083 | . 2 ⊢ (card‘𝐴) ∈ On | |
2 | cardid2 9092 | . . . 4 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
3 | domen2 8372 | . . . 4 ⊢ ((card‘𝐴) ≈ 𝐴 → (𝐵 ≼ (card‘𝐴) ↔ 𝐵 ≼ 𝐴)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ dom card → (𝐵 ≼ (card‘𝐴) ↔ 𝐵 ≼ 𝐴)) |
5 | 4 | biimpar 471 | . 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ≼ (card‘𝐴)) |
6 | ondomen 9173 | . 2 ⊢ (((card‘𝐴) ∈ On ∧ 𝐵 ≼ (card‘𝐴)) → 𝐵 ∈ dom card) | |
7 | 1, 5, 6 | sylancr 581 | 1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2164 class class class wbr 4873 dom cdm 5342 Oncon0 5963 ‘cfv 6123 ≈ cen 8219 ≼ cdom 8220 cardccrd 9074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-wrecs 7672 df-recs 7734 df-er 8009 df-en 8223 df-dom 8224 df-card 9078 |
This theorem is referenced by: ssnum 9175 indcardi 9177 fonum 9194 infpwfien 9198 inffien 9199 unnum 9337 infdif 9346 infxpabs 9349 infunsdom1 9350 infunsdom 9351 infmap2 9355 gchac 9818 grothac 9967 mbfimaopnlem 23821 ttac 38439 isnumbasgrplem2 38510 |
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