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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 9375 | . 2 ⊢ (card‘𝐴) ∈ On | |
2 | cardid2 9384 | . . . 4 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
3 | domen2 8662 | . . . 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 480 | . 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ≼ (card‘𝐴)) |
6 | ondomen 9465 | . 2 ⊢ (((card‘𝐴) ∈ On ∧ 𝐵 ≼ (card‘𝐴)) → 𝐵 ∈ dom card) | |
7 | 1, 5, 6 | sylancr 589 | 1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5068 dom cdm 5557 Oncon0 6193 ‘cfv 6357 ≈ cen 8508 ≼ cdom 8509 cardccrd 9366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-wrecs 7949 df-recs 8010 df-er 8291 df-en 8512 df-dom 8513 df-card 9370 |
This theorem is referenced by: ssnum 9467 indcardi 9469 fonum 9486 infpwfien 9490 inffien 9491 unnum 9624 infdif 9633 infxpabs 9636 infunsdom1 9637 infunsdom 9638 infmap2 9642 gchac 10105 grothac 10254 mbfimaopnlem 24258 ttac 39640 isnumbasgrplem2 39711 |
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