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Theorem gruixp 10496

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 10486	. . 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 10495	. . 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 8674	. . 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 10483	. 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 2108 ∀wral 3063 ⊆ wss 3883 ∪ ciun 4921 (class class class)co 7255 ↑_m cmap 8573 Xcixp 8643 Univcgru 10477

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-map 8575 df-pm 8576 df-ixp 8644 df-gru 10478

This theorem is referenced by: (None)

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