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Theorem gruixp 10849

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 1137	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → 𝑈 ∈ Univ)
2		gruiun 10839	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
3		simp2 1138	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → 𝐴 ∈ 𝑈)
4		grumap 10848	. . 3 ⊢ ((𝑈 ∈ Univ ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ∈ 𝑈)
5	1, 2, 3, 4	syl3anc 1373	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ∈ 𝑈)
6		ixpssmapg 8968	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴))
7	6	3ad2ant3 1136	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴))
8		gruss 10836	. 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 1373	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → X𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ∪ ciun 4991 (class class class)co 7431 ↑_m cmap 8866 Xcixp 8937 Univcgru 10830

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-pm 8869 df-ixp 8938 df-gru 10831

This theorem is referenced by: (None)

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