funfv1st2nd - Metamath Proof Explorer

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Theorem funfv1st2nd 8085

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.)

**Assertion**
Ref	Expression
funfv1st2nd	⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋))

**Proof of Theorem funfv1st2nd**
Step	Hyp	Ref	Expression
1		funrel 6579	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
2		1st2nd 8077	. . 3 ⊢ ((Rel 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
3	1, 2	sylan 592	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
4		eleq1 2852	. . . . 5 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝑋 ∈ 𝐹 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹))
5	4	adantl 487	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐹 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹))
6		funopfv 6960	. . . . 5 ⊢ (Fun 𝐹 → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹 → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
7	6	adantr 486	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹 → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
8	5, 7	sylbid 243	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐹 → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
9	8	impancom 457	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
10	3, 9	mpd 16	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ⟨cop 4838 Rel wrel 5661 Fun wfun 6555 ‘cfv 6561 1^st c1st 8026 2^nd c2nd 8027

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pr 5396 ax-un 7778

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-iota 6515 df-fun 6563 df-fv 6569 df-1st 8028 df-2nd 8029

This theorem is used by: gsumhashmul 33704 satffunlem 36346 satffunlem1lem1 36347 satffunlem2lem1 36349

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