Theorem cardiun 9413

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 7664	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ∈ V)
2		vex 3499	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3		eqeq1 2827	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = (card‘𝐵) ↔ 𝑦 = (card‘𝐵)))
4	3	rexbidv 3299	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵)))
5	2, 4	elab 3669	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ↔ ∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵))
6		cardidm 9390	. . . . . . . . . 10 ⊢ (card‘(card‘𝐵)) = (card‘𝐵)
7		fveq2 6672	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐵) → (card‘𝑦) = (card‘(card‘𝐵)))
8		id 22	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐵) → 𝑦 = (card‘𝐵))
9	6, 7, 8	3eqtr4a 2884	. . . . . . . . 9 ⊢ (𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
10	9	rexlimivw 3284	. . . . . . . 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 3150	. . . . . 6 ⊢ ∀𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦
13		carduni 9412	. . . . . 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 6685	. . . . . . 7 ⊢ (card‘𝐵) ∈ V
16	15	dfiun2 4960	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}
17	16	fveq2i 6675	. . . . 5 ⊢ (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = (card‘∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)})
18	14, 17, 16	3eqtr4g 2883	. . . 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 4940	. . . . 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 6676	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = (card‘∪ 𝑥 ∈ 𝐴 𝐵))
23	19, 22, 21	3eqtr3d 2866	. 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 1537   ∈ wcel 2114  {cab 2801  ∀wral 3140  ∃wrex 3141  Vcvv 3496  ∪ cuni 4840  ∪ ciun 4921  ‘cfv 6357  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-ord 6196  df-on 6197  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-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-card 9370

This theorem is referenced by:  alephcard  9498