difmap - Mathbox for Glauco Siliprandi

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Theorem difmap 45208

Description: Difference of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

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

**Assertion**
Ref	Expression
difmap	⊢ (𝜑 → ((𝐴 ∖ 𝐵) ↑_m 𝐶) ⊆ ((𝐴 ↑_m 𝐶) ∖ (𝐵 ↑_m 𝐶)))

Proof of Theorem difmap Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difmap.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		difssd 4103	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
3		mapss 8865	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → ((𝐴 ∖ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
4	1, 2, 3	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → ((𝐴 ∖ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
6		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶))
7	5, 6	sseldd 3950	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → 𝑓 ∈ (𝐴 ↑_m 𝐶))
8		difmap.n	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≠ ∅)
9		n0 4319	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐶)
10	8, 9	sylib 218	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐶)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → ∃𝑥 𝑥 ∈ 𝐶)
12		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑓:𝐶⟶𝐵) → 𝑓:𝐶⟶𝐵)
13		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑓:𝐶⟶𝐵) → 𝑥 ∈ 𝐶)
14	12, 13	ffvelcdmd 7060	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑓:𝐶⟶𝐵) → (𝑓‘𝑥) ∈ 𝐵)
15	14	adantll 714	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑓:𝐶⟶𝐵) → (𝑓‘𝑥) ∈ 𝐵)
16		elmapi 8825	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶) → 𝑓:𝐶⟶(𝐴 ∖ 𝐵))
17	16	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶) ∧ 𝑥 ∈ 𝐶) → 𝑓:𝐶⟶(𝐴 ∖ 𝐵))
18		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶) ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
19	17, 18	ffvelcdmd 7060	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶) ∧ 𝑥 ∈ 𝐶) → (𝑓‘𝑥) ∈ (𝐴 ∖ 𝐵))
20		eldifn 4098	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ (𝐴 ∖ 𝐵) → ¬ (𝑓‘𝑥) ∈ 𝐵)
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶) ∧ 𝑥 ∈ 𝐶) → ¬ (𝑓‘𝑥) ∈ 𝐵)
22	21	ad4ant23 753	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑓:𝐶⟶𝐵) → ¬ (𝑓‘𝑥) ∈ 𝐵)
23	15, 22	pm2.65da 816	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) → ¬ 𝑓:𝐶⟶𝐵)
24	23	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → (𝑥 ∈ 𝐶 → ¬ 𝑓:𝐶⟶𝐵))
25	24	exlimdv 1933	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → (∃𝑥 𝑥 ∈ 𝐶 → ¬ 𝑓:𝐶⟶𝐵))
26	11, 25	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → ¬ 𝑓:𝐶⟶𝐵)
27		difmap.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
28		difmap.v	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
29		elmapg 8815	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑍) → (𝑓 ∈ (𝐵 ↑_m 𝐶) ↔ 𝑓:𝐶⟶𝐵))
30	27, 28, 29	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑_m 𝐶) ↔ 𝑓:𝐶⟶𝐵))
31	30	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → (𝑓 ∈ (𝐵 ↑_m 𝐶) ↔ 𝑓:𝐶⟶𝐵))
32	26, 31	mtbird 325	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → ¬ 𝑓 ∈ (𝐵 ↑_m 𝐶))
33	7, 32	eldifd 3928	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → 𝑓 ∈ ((𝐴 ↑_m 𝐶) ∖ (𝐵 ↑_m 𝐶)))
34	33	ralrimiva 3126	. 2 ⊢ (𝜑 → ∀𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)𝑓 ∈ ((𝐴 ↑_m 𝐶) ∖ (𝐵 ↑_m 𝐶)))
35		dfss3 3938	. 2 ⊢ (((𝐴 ∖ 𝐵) ↑_m 𝐶) ⊆ ((𝐴 ↑_m 𝐶) ∖ (𝐵 ↑_m 𝐶)) ↔ ∀𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)𝑓 ∈ ((𝐴 ↑_m 𝐶) ∖ (𝐵 ↑_m 𝐶)))
36	34, 35	sylibr 234	1 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ↑_m 𝐶) ⊆ ((𝐴 ↑_m 𝐶) ∖ (𝐵 ↑_m 𝐶)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1779 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 ∖ cdif 3914 ⊆ wss 3917 ∅c0 4299 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ↑_m cmap 8802

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-map 8804

This theorem is referenced by: difmapsn 45213

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