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Theorem funfv1st2nd 8023

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 6534	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
2		1st2nd 8016	. . 3 ⊢ ((Rel 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
3	1, 2	sylan 589	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
4		eleq1 2849	. . . . 5 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝑋 ∈ 𝐹 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹))
5	4	adantl 485	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐹 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹))
6		funopfv 6912	. . . . 5 ⊢ (Fun 𝐹 → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹 → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
7	6	adantr 484	. . . 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 242	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐹 → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
9	8	impancom 455	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
10	3, 9	mpd 15	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ⟨cop 4587 Rel wrel 5650 Fun wfun 6511 ‘cfv 6517 1^st c1st 7964 2^nd c2nd 7965

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-iota 6473 df-fun 6519 df-fv 6525 df-1st 7966 df-2nd 7967

This theorem is referenced by: gsumhashmul 33208 satffunlem 35715 satffunlem1lem1 35716 satffunlem2lem1 35718

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