Theorem dfac8a 9940

Description: Numeration theorem: every set with a choice function on its power set is numerable. With AC, this reduces to the statement that every set is numerable. Similar to Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)

Assertion

Ref	Expression
dfac8a	⊢ (𝐴 ∈ 𝐵 → (∃ℎ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (ℎ‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

Distinct variable groups:   𝑦,ℎ,𝐴   𝐵,ℎ

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem dfac8a

Dummy variables 𝑓 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. 2 ⊢ recs((𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣)))) = recs((𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣))))
2		rneq 5885	. . . . 5 ⊢ (𝑣 = 𝑓 → ran 𝑣 = ran 𝑓)
3	2	difeq2d 4078	. . . 4 ⊢ (𝑣 = 𝑓 → (𝐴 ∖ ran 𝑣) = (𝐴 ∖ ran 𝑓))
4	3	fveq2d 6838	. . 3 ⊢ (𝑣 = 𝑓 → (ℎ‘(𝐴 ∖ ran 𝑣)) = (ℎ‘(𝐴 ∖ ran 𝑓)))
5	4	cbvmptv 5202	. 2 ⊢ (𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣))) = (𝑓 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑓)))
6	1, 5	dfac8alem 9939	1 ⊢ (𝐴 ∈ 𝐵 → (∃ℎ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (ℎ‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

Syntax hints:   → wi 4  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  Vcvv 3440   ∖ cdif 3898  ∅c0 4285  𝒫 cpw 4554   ↦ cmpt 5179  dom cdm 5624  ran crn 5625  ‘cfv 6492  recscrecs 8302  cardccrd 9847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-en 8884  df-card 9851

This theorem is referenced by:  ween  9945  acnnum  9962  dfac8  10046