ixpssmapc - Mathbox for Glauco Siliprandi

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Theorem ixpssmapc 45528

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 413	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶))
5	2, 4	ralrimi 3238	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
6		iunss 4981	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
7	5, 6	sylibr 235	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
8	1, 7	ssexd 5259	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)
9		ixpssmap2g 8872	. . 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 8834	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) → (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ⊆ (𝐶 ↑_m 𝐴))
12	1, 7, 11	syl2anc 590	. 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ⊆ (𝐶 ↑_m 𝐴))
13	10, 12	sstrd 3932	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐶 ↑_m 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 Ⅎwnf 1790 ∈ wcel 2119 ∀wral 3054 Vcvv 3432 ⊆ wss 3890 ∪ ciun 4928 (class class class)co 7363 ↑_m cmap 8770 Xcixp 8842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-map 8772 df-ixp 8843

This theorem is referenced by: ioorrnopnlem 46754

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