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Theorem funelss 8035

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 6564	. . . . . 6 ⊢ (Fun 𝐴 → Rel 𝐴)
2		1st2nd 8027	. . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
3	1, 2	sylan 578	. . . . 5 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
4		simpl1l 1222	. . . . . . . . . 10 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → Fun 𝐴)
5		simpl3 1191	. . . . . . . . . 10 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → 𝐵 ⊆ 𝐴)
6		simpr 483	. . . . . . . . . 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 6911	. . . . . . . . . 10 ⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ (1^st ‘𝑋) ∈ dom 𝐵) → (𝐴‘(1^st ‘𝑋)) = (𝐵‘(1^st ‘𝑋)))
8	4, 5, 6, 7	syl3anc 1369	. . . . . . . . 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 2819	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝑋 ∈ 𝐴 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴))
10	9	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐴 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐴 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴))
11		funopfv 6942	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐴 → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐴 → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
12	11	adantr 479	. . . . . . . . . . . . . 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 239	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐴 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐴 → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
14	13	impancom 450	. . . . . . . . . . . 12 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
15	14	imp 405	. . . . . . . . . . 11 ⊢ (((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋))
16	15	3adant3 1130	. . . . . . . . . 10 ⊢ (((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) → (𝐴‘(1^st ‘𝑋)) = (2^nd ‘𝑋))
17	16	adantr 479	. . . . . . . . 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 2772	. . . . . . . 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 6566	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
20	19	com12 32	. . . . . . . . . . . . 13 ⊢ (Fun 𝐴 → (𝐵 ⊆ 𝐴 → Fun 𝐵))
21	20	adantr 479	. . . . . . . . . . . 12 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐵 ⊆ 𝐴 → Fun 𝐵))
22	21	imp 405	. . . . . . . . . . 11 ⊢ (((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝐵 ⊆ 𝐴) → Fun 𝐵)
23	22	funfnd 6578	. . . . . . . . . 10 ⊢ (((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝐵 ⊆ 𝐴) → 𝐵 Fn dom 𝐵)
24	23	3adant2 1129	. . . . . . . . 9 ⊢ (((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) → 𝐵 Fn dom 𝐵)
25		fnopfvb 6944	. . . . . . . . 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 578	. . . . . . . 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 231	. . . . . . 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 2819	. . . . . . . . 9 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝑋 ∈ 𝐵 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐵))
29	28	3ad2ant2 1132	. . . . . . . 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 479	. . . . . . 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 256	. . . . . 6 ⊢ ((((Fun 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ 𝐵 ⊆ 𝐴) ∧ (1^st ‘𝑋) ∈ dom 𝐵) → 𝑋 ∈ 𝐵)
32	31	3exp1 1350	. . . . 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 411	. . 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 1109	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 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ⊆ wss 3947 ⟨cop 4633 dom cdm 5675 Rel wrel 5680 Fun wfun 6536 Fn wfn 6537 ‘cfv 6542 1^st c1st 7975 2^nd c2nd 7976

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-iota 6494 df-fun 6544 df-fn 6545 df-fv 6550 df-1st 7977 df-2nd 7978

This theorem is referenced by: funeldmdif 8036

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