Theorem xpexgALT 7962

Description: Alternate proof of xpexg 7733 requiring Replacement (ax-rep 5227) but not Power Set (ax-pow 5322). (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 5018	. . . 4 ⊢ ∪ 𝑦 ∈ 𝐵 {𝑦} = 𝐵
2	1	xpeq2i 5674	. . 3 ⊢ (𝐴 × ∪ 𝑦 ∈ 𝐵 {𝑦}) = (𝐴 × 𝐵)
3		xpiundi 5718	. . 3 ⊢ (𝐴 × ∪ 𝑦 ∈ 𝐵 {𝑦}) = ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
4	2, 3	eqtr3i 2787	. 2 ⊢ (𝐴 × 𝐵) = ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
5		id 22	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑊)
6		fconstmpt 5709	. . . . 5 ⊢ (𝐴 × {𝑦}) = (𝑥 ∈ 𝐴 ↦ 𝑦)
7		mptexg 7205	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝑦) ∈ V)
8	6, 7	eqeltrid 2866	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {𝑦}) ∈ V)
9	8	ralrimivw 3158	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
10		iunexg 7944	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
11	5, 9, 10	syl2anr 606	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V)
12	4, 11	eqeltrid 2866	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142  ∀wral 3076  Vcvv 3454  {csn 4582  ∪ ciun 4949   ↦ cmpt 5181   × cxp 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529

This theorem is referenced by: (None)