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Mirrors > Home > MPE Home > Th. List > cdafn | Structured version Visualization version GIF version |
Description: Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
cdafn | ⊢ +𝑐 Fn (V × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cda 9312 | . 2 ⊢ +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1o}))) | |
2 | vex 3417 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | snex 5131 | . . . 4 ⊢ {∅} ∈ V | |
4 | 2, 3 | xpex 7228 | . . 3 ⊢ (𝑥 × {∅}) ∈ V |
5 | vex 3417 | . . . 4 ⊢ 𝑦 ∈ V | |
6 | snex 5131 | . . . 4 ⊢ {1o} ∈ V | |
7 | 5, 6 | xpex 7228 | . . 3 ⊢ (𝑦 × {1o}) ∈ V |
8 | 4, 7 | unex 7221 | . 2 ⊢ ((𝑥 × {∅}) ∪ (𝑦 × {1o})) ∈ V |
9 | 1, 8 | fnmpt2i 7507 | 1 ⊢ +𝑐 Fn (V × V) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3414 ∪ cun 3796 ∅c0 4146 {csn 4399 × cxp 5344 Fn wfn 6122 1oc1o 7824 +𝑐 ccda 9311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-cda 9312 |
This theorem is referenced by: cda1dif 9320 cdacomen 9325 cdadom1 9330 cdainf 9336 pwcdadom 9360 |
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