Theorem dfac8a 10010

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 2763	. 2 ⊢ recs((𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣)))) = recs((𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣))))
2		rneq 5926	. . . . 5 ⊢ (𝑣 = 𝑓 → ran 𝑣 = ran 𝑓)
3	2	difeq2d 4081	. . . 4 ⊢ (𝑣 = 𝑓 → (𝐴 ∖ ran 𝑣) = (𝐴 ∖ ran 𝑓))
4	3	fveq2d 6885	. . 3 ⊢ (𝑣 = 𝑓 → (ℎ‘(𝐴 ∖ ran 𝑣)) = (ℎ‘(𝐴 ∖ ran 𝑓)))
5	4	cbvmptv 5215	. 2 ⊢ (𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣))) = (𝑓 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑓)))
6	1, 5	dfac8alem 10009	1 ⊢ (𝐴 ∈ 𝐵 → (∃ℎ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (ℎ‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

Syntax hints:   → wi 4  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  Vcvv 3455   ∖ cdif 3902  ∅c0 4286  𝒫 cpw 4562   ↦ cmpt 5192  dom cdm 5661  ran crn 5662  ‘cfv 6536  recscrecs 8353  cardccrd 9917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-en 8940  df-card 9921

This theorem is referenced by:  ween  10015  acnnum  10032  dfac8  10115