Theorem cardiun 9410

Description: The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)

Assertion

Ref	Expression
cardiun	⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem cardiun

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		abrexexg 7661	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ∈ V)
2		vex 3497	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3		eqeq1 2825	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = (card‘𝐵) ↔ 𝑦 = (card‘𝐵)))
4	3	rexbidv 3297	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵)))
5	2, 4	elab 3666	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ↔ ∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵))
6		cardidm 9387	. . . . . . . . . 10 ⊢ (card‘(card‘𝐵)) = (card‘𝐵)
7		fveq2 6669	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐵) → (card‘𝑦) = (card‘(card‘𝐵)))
8		id 22	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐵) → 𝑦 = (card‘𝐵))
9	6, 7, 8	3eqtr4a 2882	. . . . . . . . 9 ⊢ (𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
10	9	rexlimivw 3282	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
11	5, 10	sylbi 219	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} → (card‘𝑦) = 𝑦)
12	11	rgen 3148	. . . . . 6 ⊢ ∀𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦
13		carduni 9409	. . . . . 6 ⊢ ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ∈ V → (∀𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦 → (card‘∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}))
14	1, 12, 13	mpisyl 21	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (card‘∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)})
15		fvex 6682	. . . . . . 7 ⊢ (card‘𝐵) ∈ V
16	15	dfiun2 4957	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}
17	16	fveq2i 6672	. . . . 5 ⊢ (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = (card‘∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)})
18	14, 17, 16	3eqtr4g 2881	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = ∪ 𝑥 ∈ 𝐴 (card‘𝐵))
19	18	adantr 483	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = ∪ 𝑥 ∈ 𝐴 (card‘𝐵))
20		iuneq2 4937	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
21	20	adantl 484	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
22	21	fveq2d 6673	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = (card‘∪ 𝑥 ∈ 𝐴 𝐵))
23	19, 22, 21	3eqtr3d 2864	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
24	23	ex 415	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  ∃wrex 3139  Vcvv 3494  ∪ cuni 4837  ∪ ciun 4918  ‘cfv 6354  cardccrd 9363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-card 9367

This theorem is referenced by:  alephcard  9495