difmap - Mathbox for Glauco Siliprandi

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Theorem difmap 46045

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 4087	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
3		mapss 8900	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → ((𝐴 ∖ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
4	1, 2, 3	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
5	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → ((𝐴 ∖ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
6		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶))
7	5, 6	sseldd 3935	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → 𝑓 ∈ (𝐴 ↑_m 𝐶))
8		difmap.n	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≠ ∅)
9		n0 4303	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐶)
10	8, 9	sylib 221	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐶)
11	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → ∃𝑥 𝑥 ∈ 𝐶)
12		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑓:𝐶⟶𝐵) → 𝑓:𝐶⟶𝐵)
13		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑓:𝐶⟶𝐵) → 𝑥 ∈ 𝐶)
14	12, 13	ffvelcdmd 7082	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑓:𝐶⟶𝐵) → (𝑓‘𝑥) ∈ 𝐵)
15	14	adantll 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑓:𝐶⟶𝐵) → (𝑓‘𝑥) ∈ 𝐵)
16		elmapi 8852	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶) → 𝑓:𝐶⟶(𝐴 ∖ 𝐵))
17	16	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶) ∧ 𝑥 ∈ 𝐶) → 𝑓:𝐶⟶(𝐴 ∖ 𝐵))
18		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶) ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
19	17, 18	ffvelcdmd 7082	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶) ∧ 𝑥 ∈ 𝐶) → (𝑓‘𝑥) ∈ (𝐴 ∖ 𝐵))
20		eldifn 4082	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ (𝐴 ∖ 𝐵) → ¬ (𝑓‘𝑥) ∈ 𝐵)
21	19, 20	syl 18	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶) ∧ 𝑥 ∈ 𝐶) → ¬ (𝑓‘𝑥) ∈ 𝐵)
22	21	ad4ant23 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) ∧ 𝑓:𝐶⟶𝐵) → ¬ (𝑓‘𝑥) ∈ 𝐵)
23	15, 22	pm2.65da 829	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) ∧ 𝑥 ∈ 𝐶) → ¬ 𝑓:𝐶⟶𝐵)
24	23	ex 418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → (𝑥 ∈ 𝐶 → ¬ 𝑓:𝐶⟶𝐵))
25	24	exlimdv 1966	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → (∃𝑥 𝑥 ∈ 𝐶 → ¬ 𝑓:𝐶⟶𝐵))
26	11, 25	mpd 16	. . . . 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 8842	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑍) → (𝑓 ∈ (𝐵 ↑_m 𝐶) ↔ 𝑓:𝐶⟶𝐵))
30	27, 28, 29	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑_m 𝐶) ↔ 𝑓:𝐶⟶𝐵))
31	30	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → (𝑓 ∈ (𝐵 ↑_m 𝐶) ↔ 𝑓:𝐶⟶𝐵))
32	26, 31	mtbird 328	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → ¬ 𝑓 ∈ (𝐵 ↑_m 𝐶))
33	7, 32	eldifd 3913	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)) → 𝑓 ∈ ((𝐴 ↑_m 𝐶) ∖ (𝐵 ↑_m 𝐶)))
34	33	ralrimiva 3156	. 2 ⊢ (𝜑 → ∀𝑓 ∈ ((𝐴 ∖ 𝐵) ↑_m 𝐶)𝑓 ∈ ((𝐴 ↑_m 𝐶) ∖ (𝐵 ↑_m 𝐶)))
35		dfss3 3923	. 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 237	1 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ↑_m 𝐶) ⊆ ((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 ∃wex 1812 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 ∖ cdif 3899 ⊆ wss 3902 ∅c0 4282 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 ↑_m cmap 8830

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7420 df-oprab 7421 df-mpo 7422 df-1st 7990 df-2nd 7991 df-map 8832

This theorem is used by: difmapsn 46050

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