Theorem dfac8a 10036

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 2762	. 2 ⊢ recs((𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣)))) = recs((𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣))))
2		rneq 5924	. . . . 5 ⊢ (𝑣 = 𝑓 → ran 𝑣 = ran 𝑓)
3	2	difeq2d 4077	. . . 4 ⊢ (𝑣 = 𝑓 → (𝐴 ∖ ran 𝑣) = (𝐴 ∖ ran 𝑓))
4	3	fveq2d 6886	. . 3 ⊢ (𝑣 = 𝑓 → (ℎ‘(𝐴 ∖ ran 𝑣)) = (ℎ‘(𝐴 ∖ ran 𝑓)))
5	4	cbvmptv 5213	. 2 ⊢ (𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣))) = (𝑓 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑓)))
6	1, 5	dfac8alem 10035	1 ⊢ (𝐴 ∈ 𝐵 → (∃ℎ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (ℎ‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

This proof depends on syntax axioms:   → wi 4  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  Vcvv 3453   ∖ cdif 3899  ∅c0 4282  𝒫 cpw 4560   ↦ cmpt 5190  dom cdm 5659  ran crn 5660  ‘cfv 6537  recscrecs 8362  cardccrd 9943

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-en 8956  df-card 9947

This theorem is used by:  ween  10041  acnnum  10058  dfac8  10141