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Theorem cdafn 9313
 Description: Cardinal number addition is a function. (Contributed by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
cdafn +𝑐 Fn (V × V)

Proof of Theorem cdafn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef 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
42, 3xpex 7228 . . 3 (𝑥 × {∅}) ∈ V
5 vex 3417 . . . 4 𝑦 ∈ V
6 snex 5131 . . . 4 {1o} ∈ V
75, 6xpex 7228 . . 3 (𝑦 × {1o}) ∈ V
84, 7unex 7221 . 2 ((𝑥 × {∅}) ∪ (𝑦 × {1o})) ∈ V
91, 8fnmpt2i 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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