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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 6519 | . . 3 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | 1st2nd 7972 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩) | |
3 | 1, 2 | sylan 581 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩) |
4 | eleq1 2826 | . . . . 5 ⊢ (𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ → (𝑋 ∈ 𝐹 ↔ ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ ∈ 𝐹)) | |
5 | 4 | adantl 483 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩) → (𝑋 ∈ 𝐹 ↔ ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ ∈ 𝐹)) |
6 | funopfv 6895 | . . . . 5 ⊢ (Fun 𝐹 → (⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ ∈ 𝐹 → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))) | |
7 | 6 | adantr 482 | . . . 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 453 | . 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 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4593 Rel wrel 5639 Fun wfun 6491 ‘cfv 6497 1st c1st 7920 2nd c2nd 7921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fv 6505 df-1st 7922 df-2nd 7923 |
This theorem is referenced by: gsumhashmul 31901 satffunlem 33998 satffunlem1lem1 33999 satffunlem2lem1 34001 |
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