Theorem alephlim 9833

Description: Value of the aleph function at a limit ordinal. Definition 12(iii) of [Suppes] p. 91. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Assertion

Ref	Expression
alephlim	⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑥 ∈ 𝐴 (ℵ‘𝑥))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem alephlim

Step	Hyp	Ref	Expression
1		rdglim2a 8251	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (rec(har, ω)‘𝐴) = ∪ 𝑥 ∈ 𝐴 (rec(har, ω)‘𝑥))
2		df-aleph 9708	. . 3 ⊢ ℵ = rec(har, ω)
3	2	fveq1i 6767	. 2 ⊢ (ℵ‘𝐴) = (rec(har, ω)‘𝐴)
4	2	fveq1i 6767	. . . 4 ⊢ (ℵ‘𝑥) = (rec(har, ω)‘𝑥)
5	4	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → (ℵ‘𝑥) = (rec(har, ω)‘𝑥))
6	5	iuneq2i 4945	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (ℵ‘𝑥) = ∪ 𝑥 ∈ 𝐴 (rec(har, ω)‘𝑥)
7	1, 3, 6	3eqtr4g 2803	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑥 ∈ 𝐴 (ℵ‘𝑥))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∪ ciun 4924  Lim wlim 6260  ‘cfv 6426  ωcom 7702  reccrdg 8227  harchar 9302  ℵcale 9704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5208  ax-sep 5221  ax-nul 5228  ax-pr 5350  ax-un 7578

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5074  df-opab 5136  df-mpt 5157  df-tr 5191  df-id 5484  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-we 5541  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-pred 6195  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-ov 7270  df-2nd 7821  df-frecs 8084  df-wrecs 8115  df-recs 8189  df-rdg 8228  df-aleph 9708

This theorem is referenced by:  alephon  9835  alephcard  9836  alephordi  9840  cardaleph  9855  alephsing  10042  pwcfsdom  10349