Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 9525 | . 2 ⊢ (card‘𝐴) ∈ On | |
2 | cardid2 9534 | . . . 4 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
3 | domen2 8767 | . . . 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 481 | . 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ≼ (card‘𝐴)) |
6 | ondomen 9616 | . 2 ⊢ (((card‘𝐴) ∈ On ∧ 𝐵 ≼ (card‘𝐴)) → 𝐵 ∈ dom card) | |
7 | 1, 5, 6 | sylancr 590 | 1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2112 class class class wbr 5039 dom cdm 5536 Oncon0 6191 ‘cfv 6358 ≈ cen 8601 ≼ cdom 8602 cardccrd 9516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-wrecs 8025 df-recs 8086 df-er 8369 df-en 8605 df-dom 8606 df-card 9520 |
This theorem is referenced by: ssnum 9618 indcardi 9620 fonum 9637 infpwfien 9641 inffien 9642 unnum 9775 infdif 9788 infxpabs 9791 infunsdom1 9792 infunsdom 9793 infmap2 9797 gchac 10260 grothac 10409 mbfimaopnlem 24506 ttac 40502 isnumbasgrplem2 40573 |
Copyright terms: Public domain | W3C validator |