Theorem funeldmdif 8047

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 6553	. . 3 ⊢ (Fun 𝐴 → Rel 𝐴)
2		releldmdifi 8044	. . 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 3936	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5		1stdm 8039	. . . . . . . . . . . . . 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 8046	. . . . . . . . . . 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 3937	. . . . . . 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 2822	. . . . . . 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 3139	. 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 1540   ∈ wcel 2108  ∃wrex 3060   ∖ cdif 3923   ⊆ wss 3926  dom cdm 5654  Rel wrel 5659  Fun wfun 6525  ‘cfv 6531  1^st c1st 7986

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6484  df-fun 6533  df-fn 6534  df-fv 6539  df-1st 7988  df-2nd 7989

This theorem is referenced by:  satffunlem2lem2  35428