Theorem dfac8a 9458

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 2823	. 2 ⊢ recs((𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣)))) = recs((𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣))))
2		rneq 5808	. . . . 5 ⊢ (𝑣 = 𝑓 → ran 𝑣 = ran 𝑓)
3	2	difeq2d 4101	. . . 4 ⊢ (𝑣 = 𝑓 → (𝐴 ∖ ran 𝑣) = (𝐴 ∖ ran 𝑓))
4	3	fveq2d 6676	. . 3 ⊢ (𝑣 = 𝑓 → (ℎ‘(𝐴 ∖ ran 𝑣)) = (ℎ‘(𝐴 ∖ ran 𝑓)))
5	4	cbvmptv 5171	. 2 ⊢ (𝑣 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑣))) = (𝑓 ∈ V ↦ (ℎ‘(𝐴 ∖ ran 𝑓)))
6	1, 5	dfac8alem 9457	1 ⊢ (𝐴 ∈ 𝐵 → (∃ℎ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (ℎ‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

Syntax hints:   → wi 4  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  Vcvv 3496   ∖ cdif 3935  ∅c0 4293  𝒫 cpw 4541   ↦ cmpt 5148  dom cdm 5557  ran crn 5558  ‘cfv 6357  recscrecs 8009  cardccrd 9366

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-wrecs 7949  df-recs 8010  df-en 8512  df-card 9370

This theorem is referenced by:  ween  9463  acnnum  9480  dfac8  9563