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Mirrors > Home > MPE Home > Th. List > numwdom | Structured version Visualization version GIF version |
Description: A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
numwdom | ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼* 𝐴) → 𝐵 ∈ dom card) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brwdomi 8885 | . 2 ⊢ (𝐵 ≼* 𝐴 → (𝐵 = ∅ ∨ ∃𝑓 𝑓:𝐴–onto→𝐵)) | |
2 | simpr 485 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 = ∅) → 𝐵 = ∅) | |
3 | 0fin 8599 | . . . . 5 ⊢ ∅ ∈ Fin | |
4 | finnum 9230 | . . . . 5 ⊢ (∅ ∈ Fin → ∅ ∈ dom card) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ∅ ∈ dom card |
6 | 2, 5 | syl6eqel 2893 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 = ∅) → 𝐵 ∈ dom card) |
7 | fonum 9337 | . . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝑓:𝐴–onto→𝐵) → 𝐵 ∈ dom card) | |
8 | 7 | ex 413 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝑓:𝐴–onto→𝐵 → 𝐵 ∈ dom card)) |
9 | 8 | exlimdv 1915 | . . . 4 ⊢ (𝐴 ∈ dom card → (∃𝑓 𝑓:𝐴–onto→𝐵 → 𝐵 ∈ dom card)) |
10 | 9 | imp 407 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ∃𝑓 𝑓:𝐴–onto→𝐵) → 𝐵 ∈ dom card) |
11 | 6, 10 | jaodan 952 | . 2 ⊢ ((𝐴 ∈ dom card ∧ (𝐵 = ∅ ∨ ∃𝑓 𝑓:𝐴–onto→𝐵)) → 𝐵 ∈ dom card) |
12 | 1, 11 | sylan2 592 | 1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼* 𝐴) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 842 = wceq 1525 ∃wex 1765 ∈ wcel 2083 ∅c0 4217 class class class wbr 4968 dom cdm 5450 –onto→wfo 6230 Fincfn 8364 ≼* cwdom 8874 cardccrd 9217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-fin 8368 df-wdom 8876 df-card 9221 df-acn 9224 |
This theorem is referenced by: ptcmplem2 22349 |
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