Theorem funeldmdif 8033

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 6542	. . 3 ⊢ (Fun 𝐴 → Rel 𝐴)
2		releldmdifi 8030	. . 3 ⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))
3	1, 2	sylan 591	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))
4		eldif 3917	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5		1stdm 8025	. . . . . . . . . . . . . 14 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴)
6	5	ex 417	. . . . . . . . . . . . 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 485	. . . . . . . . . . 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 485	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴))
11	10	impcom 412	. . . . . . . 8 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → (1st ‘𝑥) ∈ dom 𝐴)
12		funelss 8032	. . . . . . . . . . 11 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((1st ‘𝑥) ∈ dom 𝐵 → 𝑥 ∈ 𝐵))
13	12	3expa 1134	. . . . . . . . . 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 459	. . . . . . . 8 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → ¬ (1st ‘𝑥) ∈ dom 𝐵)
16	11, 15	eldifd 3918	. . . . . . 7 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) → (1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
17	16	3adant3 1148	. . . . . 6 ⊢ (((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (1st ‘𝑥) = 𝐶) → (1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
18		eleq1 2853	. . . . . . 7 ⊢ ((1st ‘𝑥) = 𝐶 → ((1st ‘𝑥) ∈ (dom 𝐴 ∖ dom 𝐵) ↔ 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
19	18	3ad2ant3 1151	. . . . . 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 1135	. . . 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 3164	. 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 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∃wrex 3089   ∖ cdif 3904   ⊆ wss 3907  dom cdm 5652  Rel wrel 5657  Fun wfun 6519  ‘cfv 6525  1^st c1st 7972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-fv 6533  df-1st 7974  df-2nd 7975

This theorem is referenced by:  satffunlem2lem2  35769