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Theorem cdafn 8938
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 8937 . 2 +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})))
2 vex 3189 . . . 4 𝑥 ∈ V
3 snex 4871 . . . 4 {∅} ∈ V
42, 3xpex 6918 . . 3 (𝑥 × {∅}) ∈ V
5 vex 3189 . . . 4 𝑦 ∈ V
6 snex 4871 . . . 4 {1𝑜} ∈ V
75, 6xpex 6918 . . 3 (𝑦 × {1𝑜}) ∈ V
84, 7unex 6912 . 2 ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})) ∈ V
91, 8fnmpt2i 7187 1 +𝑐 Fn (V × V)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3186  cun 3554  c0 3893  {csn 4150   × cxp 5074   Fn wfn 5844  1𝑜c1o 7501   +𝑐 ccda 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fv 5857  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-cda 8937
This theorem is referenced by:  cda1dif  8945  cdacomen  8950  cdadom1  8955  cdainf  8961  pwcdadom  8985
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