Theorem funeldmdif 8087

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 6579	. . 3 ⊢ (Fun 𝐴 → Rel 𝐴)
2		releldmdifi 8084	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))
3	1, 2	sylan 592	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))
4		eldif 4052	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5		1stdm 8078	. . . . . . . . . . . . . 14 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴)
6	5	ex 418	. . . . . . . . . . . . 13 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 → (1st ‘𝑥) ∈ dom 𝐴))
7	1, 6	syl 18	. . . . . . . . . . . 12 ⊢ (Fun 𝐴 → (𝑥 ∈ 𝐴 → (1st ‘𝑥) ∈ dom 𝐴))
8	7	adantr 486	. . . . . . . . . . 11 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ 𝐴 → (1st ‘𝑥) ∈ dom 𝐴))
9	8	com12 33	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴))
10	9	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴))
11	10	impcom 413	. . . . . . . 8 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → (1st ‘𝑥) ∈ dom 𝐴)
12		funelss 8086	. . . . . . . . . . 11 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((1st ‘𝑥) ∈ dom 𝐵 → 𝑥 ∈ 𝐵))
13	12	3expa 1136	. . . . . . . . . 10 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((1st ‘𝑥) ∈ dom 𝐵 → 𝑥 ∈ 𝐵))
14	13	con3d 153	. . . . . . . . 9 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ 𝐵 → ¬ (1st ‘𝑥) ∈ dom 𝐵))
15	14	impr 460	. . . . . . . 8 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → ¬ (1st ‘𝑥) ∈ dom 𝐵)
16	11, 15	eldifd 4053	. . . . . . 7 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → (1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
17	16	3adant3 1150	. . . . . 6 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (1st ‘𝑥) = 𝐶) → (1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
18		eleq1 2852	. . . . . . 7 ⊢ ((1st ‘𝑥) = 𝐶 → ((1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵) ↔ 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
19	18	3ad2ant3 1153	. . . . . 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 1137	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ((1st ‘𝑥) = 𝐶 → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))))
22	4, 21	biimtrid 245	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1st ‘𝑥) = 𝐶 → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))))
23	22	rexlimdv 3165	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900   ∖ cdif 4048  dom cdm 5656  Rel wrel 5661  Fun wfun 6555  ‘cfv 6561  1^st c1st 8026

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-iota 6515  df-fun 6563  df-fn 6564  df-fv 6569  df-1st 8028  df-2nd 8029

This theorem is used by:  satffunlem2lem2  36351