Theorem alephlim 9964

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 8358	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (rec(har, ω)‘𝐴) = ∪ 𝑥 ∈ 𝐴 (rec(har, ω)‘𝑥))
2		df-aleph 9839	. . 3 ⊢ ℵ = rec(har, ω)
3	2	fveq1i 6829	. 2 ⊢ (ℵ‘𝐴) = (rec(har, ω)‘𝐴)
4	2	fveq1i 6829	. . . 4 ⊢ (ℵ‘𝑥) = (rec(har, ω)‘𝑥)
5	4	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → (ℵ‘𝑥) = (rec(har, ω)‘𝑥))
6	5	iuneq2i 4963	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (ℵ‘𝑥) = ∪ 𝑥 ∈ 𝐴 (rec(har, ω)‘𝑥)
7	1, 3, 6	3eqtr4g 2791	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑥 ∈ 𝐴 (ℵ‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∪ ciun 4941  Lim wlim 6313  ‘cfv 6487  ωcom 7802  reccrdg 8334  harchar 9448  ℵcale 9835

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-aleph 9839

This theorem is referenced by:  alephon  9966  alephcard  9967  alephordi  9971  cardaleph  9986  alephsing  10173  pwcfsdom  10480