mapssbi - Mathbox for Glauco Siliprandi

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Theorem mapssbi 45970

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 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ 𝑊)
3		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
4		mapss 8896	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
5	2, 3, 4	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
6	5	ex 418	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))
7		simplr 781	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ ¬ 𝐴 ⊆ 𝐵) → (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
8		nssrex 4005	. . . . . . 7 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
9	8	bilani 510	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
10		fconst6g 6774	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → (𝐶 × {𝑥}):𝐶⟶𝐴)
11	10	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 × {𝑥}):𝐶⟶𝐴)
12		mapssbi.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13		mapssbi.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
14		elmapg 8845	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑍) → ((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
15	12, 13, 14	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
16	15	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ↔ (𝐶 × {𝑥}):𝐶⟶𝐴))
17	11, 16	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶))
18	17	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶))
19	13	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝐶 ∈ 𝑍)
20	1	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝐵 ∈ 𝑊)
21		mapssbi.n	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ≠ ∅)
22	21	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝐶 ≠ ∅)
23		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶))
24	19, 20, 22, 23	snelmap 45843	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝑥 ∈ 𝐵)
25	24	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → 𝑥 ∈ 𝐵)
26		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → ¬ 𝑥 ∈ 𝐵)
27	25, 26	pm2.65da 829	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶))
28	27	3adant2 1149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶))
29		nelss 4006	. . . . . . . . . 10 ⊢ (((𝐶 × {𝑥}) ∈ (𝐴 ↑_m 𝐶) ∧ ¬ (𝐶 × {𝑥}) ∈ (𝐵 ↑_m 𝐶)) → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
30	18, 28, 29	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
31	30	3exp 1137	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))))
32	31	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ 𝐵 → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))))
33	32	rexlimdv 3167	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))
34	9, 33	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
35	34	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
36	7, 35	condan 830	. . 3 ⊢ ((𝜑 ∧ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)) → 𝐴 ⊆ 𝐵)
37	36	ex 418	. 2 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶) → 𝐴 ⊆ 𝐵))
38	6, 37	impbid 215	1 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ (𝐴 ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 ∈ wcel 2146 ≠ wne 2961 ∃wrex 3092 ⊆ wss 3908 ∅c0 4289 {csn 4594 × cxp 5664 ⟶wf 6539 (class class class)co 7423 ↑_m cmap 8833

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 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 6499 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7426 df-oprab 7427 df-mpo 7428 df-1st 7995 df-2nd 7996 df-map 8835

This theorem is used by: (None)

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