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Theorem gruixp 10794

Description: A Grothendieck universe contains indexed cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)

**Assertion**
Ref	Expression
gruixp	⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → X𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)

Distinct variable groups: 𝑥,𝑈 𝑥,𝐴 Allowed substitution hint: 𝐵(𝑥)

**Proof of Theorem gruixp**
Step	Hyp	Ref	Expression
1		simp1 1152	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → 𝑈 ∈ Univ)
2		gruiun 10784	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
3		simp2 1153	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → 𝐴 ∈ 𝑈)
4		grumap 10793	. . 3 ⊢ ((𝑈 ∈ Univ ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ∈ 𝑈)
5	1, 2, 3, 4	syl3anc 1396	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ∈ 𝑈)
6		ixpssmapg 8926	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴))
7	6	3ad2ant3 1151	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴))
8		gruss 10781	. 2 ⊢ ((𝑈 ∈ Univ ∧ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ∈ 𝑈 ∧ X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴)) → X𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
9	1, 5, 7, 8	syl3anc 1396	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → X𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1101 ∈ wcel 2149 ∀wral 3085 ⊆ wss 3913 ∪ ciun 4960 (class class class)co 7411 ↑_m cmap 8824 Xcixp 8895 Univcgru 10775

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-pow 5337 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-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-pw 4569 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-tr 5223 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-fv 6545 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7986 df-2nd 7987 df-map 8826 df-pm 8827 df-ixp 8896 df-gru 10776

This theorem is referenced by: (None)

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