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Theorem funelss 8088

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 6595	. . . . . 6 ⊢ (Fun 𝐴 → Rel 𝐴)
2		1st2nd 8080	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
3	1, 2	sylan 579	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
4		simpl1l 1224	. . . . . . . . . 10 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → Fun 𝐴)
5		simpl3 1193	. . . . . . . . . 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 6941	. . . . . . . . . 10 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ (1^st ‘𝑋) ∈ dom 𝐵) → (𝐴‘(1^st ‘𝑋)) = (𝐵‘(1^st ‘𝑋)))
8	4, 5, 6, 7	syl3anc 1371	. . . . . . . . 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 2832	. . . . . . . . . . . . . . 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 6972	. . . . . . . . . . . . . . 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 2782	. . . . . . . 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 6597	. . . . . . . . . . . . . 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 6609	. . . . . . . . . 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 6974	. . . . . . . . 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 579	. . . . . . . 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 2832	. . . . . . . . 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 1352	. . . . 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 1111	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 1087 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 ⟨cop 4654 dom cdm 5700 Rel wrel 5705 Fun wfun 6567 Fn wfn 6568 ‘cfv 6573 1^st c1st 8028 2^nd c2nd 8029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-iota 6525 df-fun 6575 df-fn 6576 df-fv 6581 df-1st 8030 df-2nd 8031

This theorem is referenced by: funeldmdif 8089

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