mapssbi - Mathbox for Glauco Siliprandi

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Theorem mapssbi 44397

Description: Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
mapssbi.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
mapssbi.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
mapssbi.c	⊢ (𝜑 → 𝐶 ∈ 𝑍)
mapssbi.n	⊢ (𝜑 → 𝐶 ≠ ∅)

**Assertion**
Ref	Expression
mapssbi	⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))

Proof of Theorem mapssbi Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapssbi.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ 𝑊)
3		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
4		mapss 8879	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
5	2, 3, 4	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
6	5	ex 412	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))
7		simplr 766	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ ¬ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
8		nssrex 44263	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
9	8	biimpi 215	. . . . . . 7 ⊢ (¬ 𝐴 ⊆ 𝐵 → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
10	9	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
11		fconst6g 6770	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → (𝐶 × {𝑥}):𝐶⟶𝐴)
12	11	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 × {𝑥}):𝐶⟶𝐴)
13		mapssbi.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
14		mapssbi.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
15		elmapg 8829	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑍) → ((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
16	13, 14, 15	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
17	16	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
18	12, 17	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶))
19	18	3adant3 1129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶))
20	14	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝐶 ∈ 𝑍)
21	1	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝐵 ∈ 𝑊)
22		mapssbi.n	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ≠ ∅)
23	22	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝐶 ≠ ∅)
24		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶))
25	20, 21, 23, 24	snelmap 44259	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝑥 ∈ 𝐵)
26	25	adantlr 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝑥 ∈ 𝐵)
27		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → ¬ 𝑥 ∈ 𝐵)
28	26, 27	pm2.65da 814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶))
29	28	3adant2 1128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶))
30		nelss 4039	. . . . . . . . . 10 ⊢ (((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ∧ ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
31	19, 29, 30	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
32	31	3exp 1116	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))))
33	32	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))))
34	33	rexlimdv 3145	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))
35	10, 34	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
36	35	adantlr 712	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
37	7, 36	condan 815	. . 3 ⊢ ((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) → 𝐴 ⊆ 𝐵)
38	37	ex 412	. 2 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶) → 𝐴 ⊆ 𝐵))
39	6, 38	impbid 211	1 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2932 ∃wrex 3062 ⊆ wss 3940 ∅c0 4314 {csn 4620 × cxp 5664 ⟶wf 6529 (class class class)co 7401 ↑_m cmap 8816

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-map 8818

This theorem is referenced by: (None)

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