Theorem xpexgALT 7664

Description: Alternate proof of xpexg 7453 requiring Replacement (ax-rep 5154) but not Power Set (ax-pow 5231). (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 4947	. . . 4 ⊢ ∪ 𝑦 ∈ 𝐵 {𝑦} = 𝐵
2	1	xpeq2i 5546	. . 3 ⊢ (𝐴 × ∪ 𝑦 ∈ 𝐵 {𝑦}) = (𝐴 × 𝐵)
3		xpiundi 5586	. . 3 ⊢ (𝐴 × ∪ 𝑦 ∈ 𝐵 {𝑦}) = ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
4	2, 3	eqtr3i 2823	. 2 ⊢ (𝐴 × 𝐵) = ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
5		id 22	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑊)
6		fconstmpt 5578	. . . . 5 ⊢ (𝐴 × {𝑦}) = (𝑥 ∈ 𝐴 ↦ 𝑦)
7		mptexg 6961	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝑦) ∈ V)
8	6, 7	eqeltrid 2894	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {𝑦}) ∈ V)
9	8	ralrimivw 3150	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
10		iunexg 7646	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
11	5, 9, 10	syl2anr 599	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
12	4, 11	eqeltrid 2894	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  {csn 4525  ∪ ciun 4881   ↦ cmpt 5110   × cxp 5517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332

This theorem is referenced by: (None)