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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 6365 | . . 3 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | 1st2nd 7731 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) | |
3 | 1, 2 | sylan 582 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) |
4 | eleq1 2899 | . . . . 5 ⊢ (𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 → (𝑋 ∈ 𝐹 ↔ 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹)) | |
5 | 4 | adantl 484 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) → (𝑋 ∈ 𝐹 ↔ 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹)) |
6 | funopfv 6710 | . . . . 5 ⊢ (Fun 𝐹 → (〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹 → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) → (〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∈ 𝐹 → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))) |
8 | 5, 7 | sylbid 242 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉) → (𝑋 ∈ 𝐹 → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))) |
9 | 8 | impancom 454 | . 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 〈cop 4566 Rel wrel 5553 Fun wfun 6342 ‘cfv 6348 1st c1st 7680 2nd c2nd 7681 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7682 df-2nd 7683 |
This theorem is referenced by: satffunlem 32669 satffunlem1lem1 32670 satffunlem2lem1 32672 |
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