Theorem dfac8a 9441

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 2798	. 2 ⊢ recs((𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣)))) = recs((𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣))))
2		rneq 5770	. . . . 5 ⊢ (𝑣 = 𝑓 → ran 𝑣 = ran 𝑓)
3	2	difeq2d 4050	. . . 4 ⊢ (𝑣 = 𝑓 → (𝐴 ∖ ran 𝑣) = (𝐴 ∖ ran 𝑓))
4	3	fveq2d 6649	. . 3 ⊢ (𝑣 = 𝑓 → (ℎ‘(𝐴 ∖ ran 𝑣)) = (ℎ‘(𝐴 ∖ ran 𝑓)))
5	4	cbvmptv 5133	. 2 ⊢ (𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣))) = (𝑓 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑓)))
6	1, 5	dfac8alem 9440	1 ⊢ (𝐴 ∈ 𝐵 → (∃ℎ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (ℎ‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

Syntax hints:   → wi 4  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  Vcvv 3441   ∖ cdif 3878  ∅c0 4243  𝒫 cpw 4497   ↦ cmpt 5110  dom cdm 5519  ran crn 5520  ‘cfv 6324  recscrecs 7990  cardccrd 9348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-wrecs 7930  df-recs 7991  df-en 8493  df-card 9352

This theorem is referenced by:  ween  9446  acnnum  9463  dfac8  9546