Theorem xpexgALT 7978

Description: Alternate proof of xpexg 7749 requiring Replacement (ax-rep 5242) but not Power Set (ax-pow 5337). (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
xpexgALT	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)

Proof of Theorem xpexgALT

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

Step	Hyp	Ref	Expression
1		iunid 5029	. . . 4 ⊢ ∪ 𝑦 ∈ 𝐵 {𝑦} = 𝐵
2	1	xpeq2i 5689	. . 3 ⊢ (𝐴 × ∪ 𝑦 ∈ 𝐵 {𝑦}) = (𝐴 × 𝐵)
3		xpiundi 5733	. . 3 ⊢ (𝐴 × ∪ 𝑦 ∈ 𝐵 {𝑦}) = ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
4	2, 3	eqtr3i 2794	. 2 ⊢ (𝐴 × 𝐵) = ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
5		id 23	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑊)
6		fconstmpt 5724	. . . . 5 ⊢ (𝐴 × {𝑦}) = (𝑥 ∈ 𝐴 ↦ 𝑦)
7		mptexg 7220	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝑦) ∈ V)
8	6, 7	eqeltrid 2873	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {𝑦}) ∈ V)
9	8	ralrimivw 3167	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
10		iunexg 7960	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
11	5, 9, 10	syl2anr 608	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
12	4, 11	eqeltrid 2873	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149  ∀wral 3085  Vcvv 3463  {csn 4594  ∪ ciun 4960   ↦ cmpt 5196   × cxp 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by: (None)