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.

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)

Syntax hints:  Vcvv 3414   ∪ cun 3796  ∅c0 4146  {csn 4399   × cxp 5344   Fn wfn 6122  1_oc1o 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