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Mirrors > Home > MPE Home > Th. List > cardf | Structured version Visualization version GIF version |
Description: The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Ref | Expression |
---|---|
cardf | ⊢ card:V⟶On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardf2 9559 | . 2 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On | |
2 | 1 | fdmi 6557 | . . . 4 ⊢ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} |
3 | cardeqv 10083 | . . . 4 ⊢ dom card = V | |
4 | 2, 3 | eqtr3i 2767 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} = V |
5 | 4 | feq2i 6537 | . 2 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On ↔ card:V⟶On) |
6 | 1, 5 | mpbi 233 | 1 ⊢ card:V⟶On |
Colors of variables: wff setvar class |
Syntax hints: {cab 2714 ∃wrex 3062 Vcvv 3408 class class class wbr 5053 dom cdm 5551 Oncon0 6213 ⟶wf 6376 ≈ cen 8623 cardccrd 9551 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-ac2 10077 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-wrecs 8047 df-recs 8108 df-en 8627 df-card 9555 df-ac 9730 |
This theorem is referenced by: inar1 10389 |
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