Theorem funeldmdif 7729

Description: Two ways of expressing membership in the difference of domains of two nested functions. (Contributed by AV, 27-Oct-2023.)

Assertion

Ref	Expression
funeldmdif	⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem funeldmdif

Step	Hyp	Ref	Expression
1		funrel 6341	. . 3 ⊢ (Fun 𝐴 → Rel 𝐴)
2		releldmdifi 7726	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))
3	1, 2	sylan 583	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))
4		eldif 3891	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5		1stdm 7721	. . . . . . . . . . . . . 14 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴)
6	5	ex 416	. . . . . . . . . . . . 13 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 → (1st ‘𝑥) ∈ dom 𝐴))
7	1, 6	syl 17	. . . . . . . . . . . 12 ⊢ (Fun 𝐴 → (𝑥 ∈ 𝐴 → (1st ‘𝑥) ∈ dom 𝐴))
8	7	adantr 484	. . . . . . . . . . 11 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ 𝐴 → (1st ‘𝑥) ∈ dom 𝐴))
9	8	com12 32	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴))
10	9	adantr 484	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴))
11	10	impcom 411	. . . . . . . 8 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → (1st ‘𝑥) ∈ dom 𝐴)
12		funelss 7728	. . . . . . . . . . 11 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((1st ‘𝑥) ∈ dom 𝐵 → 𝑥 ∈ 𝐵))
13	12	3expa 1115	. . . . . . . . . 10 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((1st ‘𝑥) ∈ dom 𝐵 → 𝑥 ∈ 𝐵))
14	13	con3d 155	. . . . . . . . 9 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ 𝐵 → ¬ (1st ‘𝑥) ∈ dom 𝐵))
15	14	impr 458	. . . . . . . 8 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → ¬ (1st ‘𝑥) ∈ dom 𝐵)
16	11, 15	eldifd 3892	. . . . . . 7 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → (1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
17	16	3adant3 1129	. . . . . 6 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (1st ‘𝑥) = 𝐶) → (1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
18		eleq1 2877	. . . . . . 7 ⊢ ((1st ‘𝑥) = 𝐶 → ((1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵) ↔ 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
19	18	3ad2ant3 1132	. . . . . 6 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (1st ‘𝑥) = 𝐶) → ((1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵) ↔ 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
20	17, 19	mpbid 235	. . . . 5 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (1st ‘𝑥) = 𝐶) → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))
21	20	3exp 1116	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ((1st ‘𝑥) = 𝐶 → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))))
22	4, 21	syl5bi 245	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1st ‘𝑥) = 𝐶 → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))))
23	22	rexlimdv 3242	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶 → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
24	3, 23	impbid 215	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   ∖ cdif 3878   ⊆ wss 3881  dom cdm 5519  Rel wrel 5524  Fun wfun 6318  ‘cfv 6324  1^st c1st 7669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-1st 7671  df-2nd 7672

This theorem is referenced by:  satffunlem2lem2  32766