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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 6585 | . . 3 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | 1st2nd 8063 | . . 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 2827 | . . . . 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 6959 | . . . . 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 1537 ∈ wcel 2106 〈cop 4637 Rel wrel 5694 Fun wfun 6557 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: gsumhashmul 33047 satffunlem 35386 satffunlem1lem1 35387 satffunlem2lem1 35389 |
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