Theorem cdafn 8874

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 8873	. 2 ⊢ +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})))
2		vex 3176	. . . 4 ⊢ 𝑥 ∈ V
3		snex 4835	. . . 4 ⊢ {∅} ∈ V
4	2, 3	xpex 6860	. . 3 ⊢ (𝑥 × {∅}) ∈ V
5		vex 3176	. . . 4 ⊢ 𝑦 ∈ V
6		snex 4835	. . . 4 ⊢ {1𝑜} ∈ V
7	5, 6	xpex 6860	. . 3 ⊢ (𝑦 × {1𝑜}) ∈ V
8	4, 7	unex 6854	. 2 ⊢ ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})) ∈ V
9	1, 8	fnmpt2i 7128	1 ⊢ +𝑐 Fn (V × V)

Syntax hints:  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125   × cxp 5036   Fn wfn 5799  1_𝑜c1o 7440   +_𝑐 ccda 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cda 8873

This theorem is referenced by:  cda1dif  8881  cdacomen  8886  cdadom1  8891  cdainf  8897  pwcdadom  8921