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Theorem funelss 7991

Description: If the first component of an element of a function is in the domain of a subset of the function, the element is a member of this subset. (Contributed by AV, 27-Oct-2023.)

**Assertion**
Ref	Expression
funelss	⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐴) → ((1^st ‘𝑋) ∈ dom 𝐵 → 𝑋 ∈ 𝐵))

**Proof of Theorem funelss**
Step	Hyp	Ref	Expression
1		funrel 6509	. . . . . 6 ⊢ (Fun 𝐴 → Rel 𝐴)
2		1st2nd 7983	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
3	1, 2	sylan 580	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
4		simpl1l 1225	. . . . . . . . . 10 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → Fun 𝐴)
5		simpl3 1194	. . . . . . . . . 10 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → 𝐵 ⊆ 𝐴)
6		simpr 484	. . . . . . . . . 10 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → (1^st ‘𝑋) ∈ dom 𝐵)
7		funssfv 6855	. . . . . . . . . 10 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ (1^st ‘𝑋) ∈ dom 𝐵) → (𝐴‘(1^st ‘𝑋)) = (𝐵‘(1^st ‘𝑋)))
8	4, 5, 6, 7	syl3anc 1373	. . . . . . . . 9 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → (𝐴‘(1^st ‘𝑋)) = (𝐵‘(1^st ‘𝑋)))
9		eleq1 2824	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝑋 ∈ 𝐴 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴))
10	9	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐴 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐴 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴))
11		funopfv 6883	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐴 → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴 → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
12	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐴 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴 → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
13	10, 12	sylbid 240	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐴 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐴 → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
14	13	impancom 451	. . . . . . . . . . . 12 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
15	14	imp 406	. . . . . . . . . . 11 ⊢ (((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋))
16	15	3adant3 1132	. . . . . . . . . 10 ⊢ (((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋))
17	16	adantr 480	. . . . . . . . 9 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋))
18	8, 17	eqtr3d 2773	. . . . . . . 8 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → (𝐵‘(1^st ‘𝑋)) = (2^nd ‘𝑋))
19		funss 6511	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
20	19	com12 32	. . . . . . . . . . . . 13 ⊢ (Fun 𝐴 → (𝐵 ⊆ 𝐴 → Fun 𝐵))
21	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐵 ⊆ 𝐴 → Fun 𝐵))
22	21	imp 406	. . . . . . . . . . 11 ⊢ (((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝐵 ⊆ 𝐴) → Fun 𝐵)
23	22	funfnd 6523	. . . . . . . . . 10 ⊢ (((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝐵 ⊆ 𝐴) → 𝐵 Fn dom 𝐵)
24	23	3adant2 1131	. . . . . . . . 9 ⊢ (((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) → 𝐵 Fn dom 𝐵)
25		fnopfvb 6885	. . . . . . . . 9 ⊢ ((𝐵 Fn dom 𝐵 ∧ (1^st ‘𝑋) ∈ dom 𝐵) → ((𝐵‘(1^st ‘𝑋)) = (2^nd ‘𝑋) ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐵))
26	24, 25	sylan 580	. . . . . . . 8 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → ((𝐵‘(1^st ‘𝑋)) = (2^nd ‘𝑋) ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐵))
27	18, 26	mpbid 232	. . . . . . 7 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐵)
28		eleq1 2824	. . . . . . . . 9 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝑋 ∈ 𝐵 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐵))
29	28	3ad2ant2 1134	. . . . . . . 8 ⊢ (((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ 𝐵 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐵))
30	29	adantr 480	. . . . . . 7 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → (𝑋 ∈ 𝐵 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐵))
31	27, 30	mpbird 257	. . . . . 6 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → 𝑋 ∈ 𝐵)
32	31	3exp1 1353	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝐵 ⊆ 𝐴 → ((1^st ‘𝑋) ∈ dom 𝐵 → 𝑋 ∈ 𝐵))))
33	3, 32	mpd 15	. . . 4 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐵 ⊆ 𝐴 → ((1^st ‘𝑋) ∈ dom 𝐵 → 𝑋 ∈ 𝐵)))
34	33	ex 412	. . 3 ⊢ (Fun 𝐴 → (𝑋 ∈ 𝐴 → (𝐵 ⊆ 𝐴 → ((1^st ‘𝑋) ∈ dom 𝐵 → 𝑋 ∈ 𝐵))))
35	34	com23 86	. 2 ⊢ (Fun 𝐴 → (𝐵 ⊆ 𝐴 → (𝑋 ∈ 𝐴 → ((1^st ‘𝑋) ∈ dom 𝐵 → 𝑋 ∈ 𝐵))))
36	35	3imp 1110	1 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐴) → ((1^st ‘𝑋) ∈ dom 𝐵 → 𝑋 ∈ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ⊆ wss 3901 ⟨cop 4586 dom cdm 5624 Rel wrel 5629 Fun wfun 6486 Fn wfn 6487 ‘cfv 6492 1^st c1st 7931 2^nd c2nd 7932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-iota 6448 df-fun 6494 df-fn 6495 df-fv 6500 df-1st 7933 df-2nd 7934

This theorem is referenced by: funeldmdif 7992

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