|   | 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 6583 | . . 3 ⊢ (Fun 𝐹 → Rel 𝐹) | |
| 2 | 1st2nd 8064 | . . 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 2829 | . . . . 5 ⊢ (𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 → (𝑋 ∈ 𝐹 ↔ 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹)) | |
| 5 | 4 | adantl 481 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) → (𝑋 ∈ 𝐹 ↔ 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹)) | 
| 6 | funopfv 6958 | . . . . 5 ⊢ (Fun 𝐹 → (〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹 → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))) | |
| 7 | 6 | adantr 480 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) → (〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹 → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))) | 
| 8 | 5, 7 | sylbid 240 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) → (𝑋 ∈ 𝐹 → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))) | 
| 9 | 8 | impancom 451 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 Rel wrel 5690 Fun wfun 6555 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 | 
| This theorem is referenced by: gsumhashmul 33064 satffunlem 35406 satffunlem1lem1 35407 satffunlem2lem1 35409 | 
| Copyright terms: Public domain | W3C validator |