![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > funfv1st2nd | Structured version Visualization version GIF version |
Description: The function value for the first component of an ordered pair is the second component of the ordered pair. (Contributed by AV, 17-Oct-2023.) |
Ref | Expression |
---|---|
funfv1st2nd | ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 6515 | . . 3 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | 1st2nd 7967 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
3 | 1, 2 | sylan 580 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) |
4 | eleq1 2825 | . . . . 5 ⊢ (𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 → (𝑋 ∈ 𝐹 ↔ 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹)) | |
5 | 4 | adantl 482 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) → (𝑋 ∈ 𝐹 ↔ 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹)) |
6 | funopfv 6891 | . . . . 5 ⊢ (Fun 𝐹 → (〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹 → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))) | |
7 | 6 | adantr 481 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) → (〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹 → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))) |
8 | 5, 7 | sylbid 239 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) → (𝑋 ∈ 𝐹 → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))) |
9 | 8 | impancom 452 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))) |
10 | 3, 9 | mpd 15 | 1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 〈cop 4590 Rel wrel 5636 Fun wfun 6487 ‘cfv 6493 1st c1st 7915 2nd c2nd 7916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6445 df-fun 6495 df-fv 6501 df-1st 7917 df-2nd 7918 |
This theorem is referenced by: gsumhashmul 31781 satffunlem 33864 satffunlem1lem1 33865 satffunlem2lem1 33867 |
Copyright terms: Public domain | W3C validator |