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Theorem gruixp 10818

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 1134	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → 𝑈 ∈ Univ)
2		gruiun 10808	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)
3		simp2 1135	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → 𝐴 ∈ 𝑈)
4		grumap 10817	. . 3 ⊢ ((𝑈 ∈ Univ ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ∈ 𝑈)
5	1, 2, 3, 4	syl3anc 1369	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ∈ 𝑈)
6		ixpssmapg 8936	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴))
7	6	3ad2ant3 1133	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴))
8		gruss 10805	. 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 1369	1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → X𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1085 ∈ wcel 2099 ∀wral 3056 ⊆ wss 3944 ∪ ciun 4991 (class class class)co 7414 ↑_m cmap 8834 Xcixp 8905 Univcgru 10799

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7985 df-2nd 7986 df-map 8836 df-pm 8837 df-ixp 8906 df-gru 10800

This theorem is referenced by: (None)

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