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Mirrors > Home > MPE Home > Th. List > fodomnum | Structured version Visualization version GIF version |
Description: A version of fodom 10518 that doesn't require the Axiom of Choice ax-ac 10454. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fodomnum | ⊢ (𝐴 ∈ dom card → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | focdmex 7942 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V)) | |
2 | 1 | com12 32 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐴 ∈ dom card → 𝐵 ∈ V)) |
3 | numacn 10044 | . . . 4 ⊢ (𝐵 ∈ V → (𝐴 ∈ dom card → 𝐴 ∈ AC 𝐵)) | |
4 | 2, 3 | syli 39 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐴 ∈ dom card → 𝐴 ∈ AC 𝐵)) |
5 | 4 | com12 32 | . 2 ⊢ (𝐴 ∈ dom card → (𝐹:𝐴–onto→𝐵 → 𝐴 ∈ AC 𝐵)) |
6 | fodomacn 10051 | . 2 ⊢ (𝐴 ∈ AC 𝐵 → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) | |
7 | 5, 6 | syli 39 | 1 ⊢ (𝐴 ∈ dom card → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3475 class class class wbr 5149 dom cdm 5677 –onto→wfo 6542 ≼ cdom 8937 cardccrd 9930 AC wacn 9933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-card 9934 df-acn 9937 |
This theorem is referenced by: fonum 10053 fodomfi2 10055 infpwfien 10057 inffien 10058 wdomnumr 10059 iunfictbso 10109 infmap2 10213 fictb 10240 cfflb 10254 cfslb2n 10263 fodomg 10517 rankcf 10772 tskuni 10778 tskurn 10784 znnen 16155 qnnen 16156 cygctb 19760 1stcrestlem 22956 2ndcctbss 22959 2ndcomap 22962 2ndcsep 22963 tx1stc 23154 tx2ndc 23155 met1stc 24030 met2ndci 24031 re2ndc 24317 uniiccdif 25095 dyadmbl 25117 opnmblALT 25120 mbfimaopnlem 25172 aannenlem3 25843 rn1st 43978 |
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