ixpssmapc - Mathbox for Glauco Siliprandi

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Theorem ixpssmapc 41356

Description: An infinite Cartesian product is a subset of set exponentiation. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
ixpssmapc.x	⊢ Ⅎ𝑥𝜑
ixpssmapc.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
ixpssmapc.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)

**Assertion**
Ref	Expression
ixpssmapc	⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐶 ↑_m 𝐴))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐶 Allowed substitution hints: 𝜑(𝑥) 𝐵(𝑥) 𝑉(𝑥)

**Proof of Theorem ixpssmapc**
Step	Hyp	Ref	Expression
1		ixpssmapc.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
2		ixpssmapc.x	. . . . . 6 ⊢ Ⅎ𝑥𝜑
3		ixpssmapc.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
4	3	ex 415	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶))
5	2, 4	ralrimi 3216	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
6		iunss 4969	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
7	5, 6	sylibr 236	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
8	1, 7	ssexd 5228	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
9		ixpssmap2g 8491	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ V → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴))
10	8, 9	syl 17	. 2 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴))
11		mapss 8453	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ⊆ (𝐶 ↑_m 𝐴))
12	1, 7, 11	syl2anc 586	. 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ⊆ (𝐶 ↑_m 𝐴))
13	10, 12	sstrd 3977	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐶 ↑_m 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 Ⅎwnf 1784 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 ∪ ciun 4919 (class class class)co 7156 ↑_m cmap 8406 Xcixp 8461

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 df-ixp 8462

This theorem is referenced by: ioorrnopnlem 42609

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