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Theorem releldmdifi 7980

Description: One way of expressing membership in the difference of domains of two nested relations. (Contributed by AV, 26-Oct-2023.)

**Assertion**
Ref	Expression
releldmdifi	⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1^st ‘𝑥) = 𝐶))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶

**Proof of Theorem releldmdifi**
Step	Hyp	Ref	Expression
1		eldif 3913	. . 3 ⊢ (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ (𝐶 ∈ dom 𝐴 ∧ ¬ 𝐶 ∈ dom 𝐵))
2		releldm2 7978	. . . . 5 ⊢ (Rel 𝐴 → (𝐶 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶))
3	2	adantr 480	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶))
4	3	anbi1d 631	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐶 ∈ dom 𝐴 ∧ ¬ 𝐶 ∈ dom 𝐵) ↔ (∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)))
5	1, 4	bitrid 283	. 2 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ (∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)))
6		simprl 770	. . . 4 ⊢ (((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶)
7		relss 5725	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ 𝐴 → (Rel 𝐴 → Rel 𝐵))
8	7	impcom 407	. . . . . . . . . . 11 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → Rel 𝐵)
9		1stdm 7975	. . . . . . . . . . 11 ⊢ ((Rel 𝐵 ∧ 𝑥 ∈ 𝐵) → (1^st ‘𝑥) ∈ dom 𝐵)
10	8, 9	sylan 580	. . . . . . . . . 10 ⊢ (((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐵) → (1^st ‘𝑥) ∈ dom 𝐵)
11		eleq1 2816	. . . . . . . . . 10 ⊢ ((1^st ‘𝑥) = 𝐶 → ((1^st ‘𝑥) ∈ dom 𝐵 ↔ 𝐶 ∈ dom 𝐵))
12	10, 11	syl5ibcom 245	. . . . . . . . 9 ⊢ (((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝑥) = 𝐶 → 𝐶 ∈ dom 𝐵))
13	12	rexlimdva 3130	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ 𝐵 (1^st ‘𝑥) = 𝐶 → 𝐶 ∈ dom 𝐵))
14	13	con3d 152	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ 𝐶 ∈ dom 𝐵 → ¬ ∃𝑥 ∈ 𝐵 (1^st ‘𝑥) = 𝐶))
15		ralnex 3055	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ¬ (1^st ‘𝑥) = 𝐶 ↔ ¬ ∃𝑥 ∈ 𝐵 (1^st ‘𝑥) = 𝐶)
16	14, 15	imbitrrdi 252	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ 𝐶 ∈ dom 𝐵 → ∀𝑥 ∈ 𝐵 ¬ (1^st ‘𝑥) = 𝐶))
17	16	adantld 490	. . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵) → ∀𝑥 ∈ 𝐵 ¬ (1^st ‘𝑥) = 𝐶))
18	17	imp 406	. . . 4 ⊢ (((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∀𝑥 ∈ 𝐵 ¬ (1^st ‘𝑥) = 𝐶)
19		rexdifi 4101	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶 ∧ ∀𝑥 ∈ 𝐵 ¬ (1^st ‘𝑥) = 𝐶) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1^st ‘𝑥) = 𝐶)
20	6, 18, 19	syl2anc 584	. . 3 ⊢ (((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1^st ‘𝑥) = 𝐶)
21	20	ex 412	. 2 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1^st ‘𝑥) = 𝐶))
22	5, 21	sylbid 240	1 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1^st ‘𝑥) = 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ∖ cdif 3900 ⊆ wss 3903 dom cdm 5619 Rel wrel 5624 ‘cfv 6482 1^st c1st 7922

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 2701 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-iota 6438 df-fun 6484 df-fv 6490 df-1st 7924 df-2nd 7925

This theorem is referenced by: funeldmdif 7983 satffunlem2lem2 35379

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