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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 9035 | . 2 ⊢ (𝐵 ≼* 𝐴 → (𝐵 = ∅ ∨ ∃𝑓 𝑓:𝐴–onto→𝐵)) | |
2 | simpr 487 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 = ∅) → 𝐵 = ∅) | |
3 | 0fin 8749 | . . . . 5 ⊢ ∅ ∈ Fin | |
4 | finnum 9380 | . . . . 5 ⊢ (∅ ∈ Fin → ∅ ∈ dom card) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ ∅ ∈ dom card |
6 | 2, 5 | eqeltrdi 2924 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 = ∅) → 𝐵 ∈ dom card) |
7 | fonum 9487 | . . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝑓:𝐴–onto→𝐵) → 𝐵 ∈ dom card) | |
8 | 7 | ex 415 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝑓:𝐴–onto→𝐵 → 𝐵 ∈ dom card)) |
9 | 8 | exlimdv 1933 | . . . 4 ⊢ (𝐴 ∈ dom card → (∃𝑓 𝑓:𝐴–onto→𝐵 → 𝐵 ∈ dom card)) |
10 | 9 | imp 409 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ∃𝑓 𝑓:𝐴–onto→𝐵) → 𝐵 ∈ dom card) |
11 | 6, 10 | jaodan 954 | . 2 ⊢ ((𝐴 ∈ dom card ∧ (𝐵 = ∅ ∨ ∃𝑓 𝑓:𝐴–onto→𝐵)) → 𝐵 ∈ dom card) |
12 | 1, 11 | sylan2 594 | 1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼* 𝐴) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ∅c0 4294 class class class wbr 5069 dom cdm 5558 –onto→wfo 6356 Fincfn 8512 ≼* cwdom 9024 cardccrd 9367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 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 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-fin 8516 df-wdom 9026 df-card 9371 df-acn 9374 |
This theorem is referenced by: ptcmplem2 22664 |
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