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Theorem funfv1st2nd 7999

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 6515	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
2		1st2nd 7992	. . 3 ⊢ ((Rel 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
3	1, 2	sylan 581	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
4		eleq1 2824	. . . . 5 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝑋 ∈ 𝐹 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹))
5	4	adantl 481	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐹 ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹))
6		funopfv 6889	. . . . 5 ⊢ (Fun 𝐹 → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ 𝐹 → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
7	6	adantr 480	. . . 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 240	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩) → (𝑋 ∈ 𝐹 → (𝐹‘(1^st ‘𝑋)) = (2^nd ‘𝑋)))
9	8	impancom 451	. 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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⟨cop 4573 Rel wrel 5636 Fun wfun 6492 ‘cfv 6498 1^st c1st 7940 2^nd c2nd 7941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6454 df-fun 6500 df-fv 6506 df-1st 7942 df-2nd 7943

This theorem is referenced by: gsumhashmul 33128 satffunlem 35583 satffunlem1lem1 35584 satffunlem2lem1 35586

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