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Theorem cdafn 9326
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 9325 . 2 +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1o})))
2 vex 3401 . . . 4 𝑥 ∈ V
3 snex 5140 . . . 4 {∅} ∈ V
42, 3xpex 7240 . . 3 (𝑥 × {∅}) ∈ V
5 vex 3401 . . . 4 𝑦 ∈ V
6 snex 5140 . . . 4 {1o} ∈ V
75, 6xpex 7240 . . 3 (𝑦 × {1o}) ∈ V
84, 7unex 7233 . 2 ((𝑥 × {∅}) ∪ (𝑦 × {1o})) ∈ V
91, 8fnmpt2i 7519 1 +𝑐 Fn (V × V)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3398  cun 3790  c0 4141  {csn 4398   × cxp 5353   Fn wfn 6130  1oc1o 7836   +𝑐 ccda 9324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-cda 9325
This theorem is referenced by:  cda1dif  9333  cdacomen  9338  cdadom1  9343  cdainf  9349  pwcdadom  9373
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