Theorem funeldmdif 8055

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 6563	. . 3 ⊢ (Fun 𝐴 → Rel 𝐴)
2		releldmdifi 8052	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))
3	1, 2	sylan 580	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))
4		eldif 3941	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5		1stdm 8047	. . . . . . . . . . . . . 14 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴)
6	5	ex 412	. . . . . . . . . . . . 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 480	. . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴))
11	10	impcom 407	. . . . . . . 8 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → (1st ‘𝑥) ∈ dom 𝐴)
12		funelss 8054	. . . . . . . . . . 11 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((1st ‘𝑥) ∈ dom 𝐵 → 𝑥 ∈ 𝐵))
13	12	3expa 1118	. . . . . . . . . 10 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((1st ‘𝑥) ∈ dom 𝐵 → 𝑥 ∈ 𝐵))
14	13	con3d 152	. . . . . . . . 9 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ 𝐵 → ¬ (1st ‘𝑥) ∈ dom 𝐵))
15	14	impr 454	. . . . . . . 8 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → ¬ (1st ‘𝑥) ∈ dom 𝐵)
16	11, 15	eldifd 3942	. . . . . . 7 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → (1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
17	16	3adant3 1132	. . . . . 6 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (1st ‘𝑥) = 𝐶) → (1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
18		eleq1 2821	. . . . . . 7 ⊢ ((1st ‘𝑥) = 𝐶 → ((1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵) ↔ 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
19	18	3ad2ant3 1135	. . . . . 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 232	. . . . 5 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (1st ‘𝑥) = 𝐶) → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))
21	20	3exp 1119	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ((1st ‘𝑥) = 𝐶 → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))))
22	4, 21	biimtrid 242	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1st ‘𝑥) = 𝐶 → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))))
23	22	rexlimdv 3140	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶 → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
24	3, 23	impbid 212	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∃wrex 3059   ∖ cdif 3928   ⊆ wss 3931  dom cdm 5665  Rel wrel 5670  Fun wfun 6535  ‘cfv 6541  1^st c1st 7994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-iota 6494  df-fun 6543  df-fn 6544  df-fv 6549  df-1st 7996  df-2nd 7997

This theorem is referenced by:  satffunlem2lem2  35386