funelss - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > funelss		Structured version Visualization version GIF version

Theorem funelss 7989

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 6503	. . . . . 6 ⊢ (Fun 𝐴 → Rel 𝐴)
2		1st2nd 7981	. . . . . 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 6847	. . . . . . . . . 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 2816	. . . . . . . . . . . . . . 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 6876	. . . . . . . . . . . . . . 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 2766	. . . . . . . 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 6505	. . . . . . . . . . . . . 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 6517	. . . . . . . . . 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 6878	. . . . . . . . 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 2816	. . . . . . . . 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 1540 ∈ wcel 2109 ⊆ wss 3905 ⟨cop 4585 dom cdm 5623 Rel wrel 5628 Fun wfun 6480 Fn wfn 6481 ‘cfv 6486 1^st c1st 7929 2^nd c2nd 7930

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pr 5374 ax-un 7675

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-iota 6442 df-fun 6488 df-fn 6489 df-fv 6494 df-1st 7931 df-2nd 7932

This theorem is referenced by: funeldmdif 7990

W3C validator