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Theorem releldmdifi 8068

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 3972	. . 3 ⊢ (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ (𝐶 ∈ dom 𝐴 ∧ ¬ 𝐶 ∈ dom 𝐵))
2		releldm2 8066	. . . . 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 771	. . . 4 ⊢ (((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∃𝑥 ∈ 𝐴 (1^st ‘𝑥) = 𝐶)
7		relss 5793	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ 𝐴 → (Rel 𝐴 → Rel 𝐵))
8	7	impcom 407	. . . . . . . . . . 11 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → Rel 𝐵)
9		1stdm 8063	. . . . . . . . . . 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 2826	. . . . . . . . . 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 3152	. . . . . . . 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 3069	. . . . . . 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 4159	. . . 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 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 ∖ cdif 3959 ⊆ wss 3962 dom cdm 5688 Rel wrel 5693 ‘cfv 6562 1^st c1st 8010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fv 6570 df-1st 8012 df-2nd 8013

This theorem is referenced by: funeldmdif 8071 satffunlem2lem2 35390

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